\section*{Introduction} \label{introduction}

\subsection{Classical setting}

\subsubsection{Multiple zeta values} 


Multiple zeta values of Euler (MZV's for short) are real positive numbers given by
	\[ \zeta(n_1,\dots,n_r)=\sum_{0<k_1<\dots<k_r} \frac{1}{k_1^{n_1} \dots k_r^{n_r}}, \quad \text{where } n_i \geq 1, n_r \geq 2. \]
Here $r$ is called the depth and $w=n_1+\dots+n_r$ is called the weight of the presentation $\zeta(n_1,\dots,n_r)$. These values covers the special values $\zeta(n)$ for $n \geq 2$ of the Riemann zeta function and have been studied intensively especially in the last three decades with important and deep connections to different branches of mathematics and physics, for example arithmetic geometry, knot theory and higher energy physics. We refer the reader to  \cite{BGF, Zag94} for more details. 

The main goal of this theory is to understand all $\mathbb Q$-linear relations among MZV's. Goncharov conjectures that all $\mathbb Q$-linear relations among MZV's can be derived from those among MZV's of the same weight. As the next step, precise conjectures formulated by Zagier \cite{Zag94} and Hoffman \cite{Hof97} predict the dimension and an explicit basis for the $\mathbb Q$-vector space $\mathcal{Z}_k$ spanned by MZV's of weight $k$ for $k \in \mathbb{N}$.

\begin{conjecture}[Zagier's conjecture]
We define a Fibonacci-like sequence of integers $d_k$ as follows. Letting $d_0=1, d_1=0$ and $d_2=1$ we define $d_k=d_{k-2}+d_{k-3}$ for $k \geq 3$. Then for $k \in \ensuremath \mathbb{N}$ we have 
	\[ \dim_{\mathbb Q} \mathcal Z_k = d_k. \]
\end{conjecture}

\begin{conjecture}[Hoffman's conjecture]
The $\mathbb Q$-vector space $\mathcal Z_k$ is generated by the basis consisting of MZV's of weight $k$ of the form $\zeta(n_1,\dots,n_r)$ with $n_i \in \{2,3\}$.
\end{conjecture}

The algebraic part of these conjectures which concerns upper bounds for $\dim_{\mathbb Q} \mathcal Z_k$ was solved by Terasoma \cite{Ter02}, Deligne-Goncharov \cite{DG05} and Brown \cite{Bro12} using the theory of mixed Tate motives. 

\begin{theorem}[Deligne-Goncharov, Terasoma]
For $k \in \ensuremath \mathbb{N}$ we have $\dim_{\mathbb Q} \mathcal Z_k \leq d_k$.
\end{theorem}

\begin{theorem}[Brown]
The $\mathbb Q$-vector space $\mathcal Z_k$ is generated by MZV's of weight $k$ of the form $\zeta(n_1,\dots,n_r)$ with $n_i \in \{2,3\}$.
\end{theorem}

Unfortunately, the transcendental part which concerns lower bounds for $\dim_{\mathbb Q} \mathcal Z_k$ is completely open. We refer the reader to \cite{BGF,Del13,Zag94} for more details and more exhaustive references.

\subsubsection{Alternating multiple zeta values} 

There exists a variant of MZV's called the alternating multiple zeta values (AMZV's for short), also known as Euler sums. They are real numbers given by
	\[ \zeta \begin{pmatrix}
	\epsilon_1 & \dots & \epsilon_r \\
	n_1 & \dots & n_r 
	\end{pmatrix}=\sum_{0<k_1<\dots<k_r} \frac{\epsilon_1^{k_1} \dots \epsilon_r^{k_r}}{k_1^{n_1} \dots k_r^{n_r}} \]
where $\epsilon_i \in \{\pm 1\}$, $n_i \in \ensuremath \mathbb{N}$ and $(n_r,\epsilon_r) \neq (1,1)$. Similar to MZV's, these values have been studied by Broadhurst, Deligne–Goncharov, Hoffman, Kaneko–Tsumura and many others due to many connections in different contexts. We refer the reader to \cite{Har21,Hof19,Zha16} for more references.

As before, it is expected that all $\mathbb Q$-linear relations among AMZV's can be derived from those among AMZV's of the same weight. In particular, it is natural to ask whether one could formulate similar conjectures to those of Zagier and Hoffman for AMZV's of fixed weight. By the work of Deligne-Goncharov \cite{DG05}, the sharp upper bounds are achieved:
\begin{theorem}[Deligne-Goncharov]
For $k \in \ensuremath \mathbb{N}$ if we denote by $\mathcal A_k$ the $\mathbb Q$-vector space spanned by AMZV's of weight $k$, then $\dim_{\mathbb Q} \mathcal A_k \leq F_{k+1}$. Here $F_n$ is the $n$-th Fibonacci number defined by $F_1=F_2=1$ and $F_{n+2}=F_{n+1}+F_n$ for all $n \geq 1$.
\end{theorem}
 
The fact that the previous upper bounds are sharp was also explained by Deligne in \cite{Del10} using a variant of a conjecture of Grothendieck. In the direction of extending Brown's theorem for AMZV's, there exist several sets of generators for $\mathcal A_k$ (see for example \cite{Cha21,Del10}). However, we mention that these generators are only linear combinations of AMZV's. 

Finally, we know nothing about non trivial lower bounds for $\dim_{\mathbb Q} \mathcal A_k$.

\subsection{Function field setting} \label{sec:function fields}
 
\subsubsection{MZV's of Thakur and analogues of Zagier-Hoffman's conjectures} \label{sec:ZagierHoffman}

By analogy between number fields and function fields, based on the pioneering work of Carlitz \cite{Car35}, Thakur \cite{Tha04} defined analogues of multiple zeta values in positive characteristic. We now need to introduce some notations. Let $A=\mathbb F_q[\theta]$ be the polynomial ring in the variable $\theta$ over a finite field $\mathbb F_q$ of $q$ elements of characteristic $p>0$. We denote by $A_+$ the set of monic polynomials in $A$.  Let $K=\mathbb F_q(\theta)$ be the fraction field of $A$ equipped with the rational point $\infty$. Let $K_\infty$ be the completion of $K$ at $\infty$ and $\mathbb{C}_\infty$ be the completion of a fixed algebraic closure $\overline K$ of $K$ at $\infty$. We denote by $v_\infty$ the discrete valuation on $K$ corresponding to the place $\infty$ normalized such that $v_\infty(\theta)=-1$, and by $\lvert\cdot\rvert_\infty= q^{-v_\infty}$ the associated absolute value on $K$.  The unique valuation of $\mathbb C_\infty$ which extends $v_\infty$ will still be denoted by $v_\infty$. 

Let $\mathbb N=\{1,2,\dots\}$ be the set of positive integers and $\mathbb Z^{\geq 0}=\{0,1,2,\dots\}$ be the set of non-negative integers. In \cite{Car35} Carlitz introduced the Carlitz zeta values $\zeta_A(n)$ for $n \in \ensuremath \mathbb{N}$ given by
	\[ \zeta_A(n) := \sum_{a \in A_+} \frac{1}{a^n} \in K_\infty \]
which are analogues of classical special zeta values in the function field setting.  For any tuple of positive integers $\mathfrak s=(s_1,\ldots,s_r) \in \ensuremath \mathbb{N}^r$, Thakur \cite{Tha04} defined the characteristic $p$ multiple zeta value (MZV for short) $\zeta_A(\mathfrak{s})$ or $\zeta_A(s_1,\ldots,s_r)$ by
\begin{equation*}
\zeta_A(\mathfrak{s}):=\sum \frac{1}{a_1^{s_1} \ldots a_r^{s_r}} \in K_\infty
\end{equation*}
where the sum runs through the set of tuples $(a_1,\ldots,a_r) \in A_+^r$ with $\deg a_1>\cdots>\deg a_r$. We call $r$ the depth of $\zeta_A(\mathfrak{s})$ and $w(\mathfrak{s})=s_1+\dots+s_r$ the weight of $\zeta_A(\mathfrak{s})$. We note that Carlitz zeta values are exactly depth one MZV's. Thakur \cite{Tha09a} showed that all the MZV's do not vanish. We refer the reader to \cite{AT90,AT09,GP21,LRT14,LRT21,Pel12,Tha04,Tha09,Tha10,Tha17,Tha20,Yu91} for more details about these objects.

As in the classical setting, the main goal of the theory is to understand all linear relations over $K$ among MZV's.  We now state analogues of Zagier-Hoffman's conjectures in positive characteristic formulated by Thakur in \cite[\S 8]{Tha17} and by Todd in \cite{Tod18}. 

For $w \in \ensuremath \mathbb{N}$ we denote by $\mathcal Z_w$ the $K$-vector space spanned by the MZV's of weight $w$. We denote by $\mathcal T_w$ the set of $\zeta_A(\mathfrak{s})$ where $\mathfrak{s}=(s_1,\ldots,s_r) \in \ensuremath \mathbb{N}^r$ of weight $w$ with $1\leq s_i\leq q$ for $1\leq i\leq r-1$ and $s_r<q$.

\begin{conjecture}[Zagier's conjecture in positive characteristic] \label{conj: dimension}
Letting 
\begin{align*}
d(w)=\begin{cases}
1 & \text{ if } w=0, \\
2^{w-1} & \text{ if } 1 \leq w \leq q-1, \\
2^{w-1}-1 & \text{ if } w=q,
\end{cases}
\end{align*}
we put $d(w)=\sum_{i=1}^q d(w-i)$ for $w>q$. Then for any $w \in \ensuremath \mathbb{N}$, we have
	\[ \dim_K \mathcal Z_w = d(w). \]
\end{conjecture}

\begin{conjecture}[Hoffman's conjecture in positive characteristic] \label{conj: basis}
A $K$-basis for $\mathcal Z_w$ is given by  $\mathcal T_w$ consisting of $\zeta_A(s_1,\ldots,s_r)$ of weight $w$ with  $s_i \leq q$ for $1 \leq i <r$, and $s_r<q$.
\end{conjecture}

In \cite{ND21} one of the authors succeeded in proving the algebraic part of these conjectures (see \cite[Theorem A]{ND21}): for all $w \in \ensuremath \mathbb{N}$, we have
	\[ \dim_K \mathcal Z_w \leq d(w). \]
This part is based on shuffle relations for MZV's due to Chen and Thakur and some operations introduced by Todd. For the transcendental part, he used the Anderson-Brownawell-Papanikolas criterion in \cite{ABP04} and proved sharp lower bounds for small weights $w \leq 2q-2$ (see \cite[Theorem D]{ND21}). It was already noted that it is very difficult to extend his method for general weights (see \cite{ND21} for more details).

\subsubsection{AMZV's in positive characteristic}

Recently, Harada \cite{Har21} introduced the alternating multiple zeta values in positive characteristic (AMZV's) as follows. Letting $\mathfrak{s}=(s_1,\dots,s_r) \in \ensuremath \mathbb{N}^n$ and $\bm{\varepsilon}=(\varepsilon_1,\dots,\varepsilon_r) \in (\mathbb F_q^\times)^n$, we define
\begin{equation*}
    \zeta_A \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  = \sum \dfrac{\varepsilon_1^{\deg a_1} \dots \varepsilon_r^{\deg a_r }}{a_1^{s_1} \dots a_r^{s_r}}  \in K_{\infty}
\end{equation*}
where the sum runs through the set of tuples $(a_1,\ldots,a_r) \in A_+^r$ with $\deg a_1>\cdots>\deg a_r$. The numbers $r$ and $w(\mathfrak{s})=s_1+\dots+s_r$ are called the depth and the weight of $\zeta_A \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}$, respectively. Harada \cite{Har21} extended basic properties of MZV's to AMZV's, i.e., non-vanishing, shuffle relations, period interpretation and linear independence. Again the main goal of this theory is to determine all linear relations over $K$ among AMZV's. It is natural to ask whether one could extend the previous work on analogues of Zagier-Hoffman's conjectures to this setting. More precisely, if for $w \in \ensuremath \mathbb{N}$ we denote by $\mathcal{AZ}_w$ the $K$-vector space spanned by the AMZV's of weight $w$, then we would like to determine the dimensions of $\mathcal{AZ}_w$ and exhibit some nice bases of these vector spaces.

\subsection{Main results}

\subsubsection{Statements of the main results}

In this manuscript we present complete answers to all the previous conjectures and problems raised in \S \ref{sec:function fields}. 

First, for all $w$ we calculate the dimension of $\mathcal{AZ}_w$ and give an explicit basis in the spirit of Hoffman. 
\begin{theoremx} \label{thm: ZagierHoffman AMZV}
We define a Fibonacci-like sequence $s(w)$ as follows. We put
\begin{equation*}
    s(w) = \begin{cases}
(q - 1) q^{w-1}& \text{if } 1 \leq w < q, \\
            (q - 1) (q^{w-1} - 1) &\text{if } w = q,
		 \end{cases}
\end{equation*}
and for $w>q$, $s(w)=(q-1)\sum \limits_{i = 1}^{q-1} s(w-i) + s(w - q)$. Then for all $w \in \ensuremath \mathbb{N}$,  
	\[ \dim_K \mathcal{AZ}_w = s(w). \]
	
Further, we can exhibit a Hoffman-like basis of $\mathcal{AZ}_w$.
\end{theoremx}

Second, we give a proof of both Conjectures \ref{conj: dimension} and \ref{conj: basis} which generalizes the previous work of the fourth author \cite{ND21}.
\begin{theoremx} \label{thm: ZagierHoffman}
For all $w \in \ensuremath \mathbb{N}$,  $\mathcal T_w$ is a $K$-basis for $\mathcal Z_w$. In particular, 
	\[ \dim_K \mathcal Z_w = d(w). \]
\end{theoremx}

We recall that analogues of Goncharov's conjectures in positive characteristic were proved in \cite{Cha14}. As a consequence, we completely determine all linear relations over $K$ among MZV's and AMZV's and settle the main goals of these theories.

\subsubsection{Ingredients of the proofs} ${}$\par

Let us emphasize here that Theorem \ref{thm: ZagierHoffman AMZV} is much harder than Theorem \ref{thm: ZagierHoffman} and that it is not enough to work inside the setting of AMZV's. On the one hand, although it is straightforward to extend the algeraic part for AMZV's following the same line in \cite[\S 2 and \S 3]{ND21}, we only obtain a weak version of Brown's theorem in this setting. More precisely, we get a set of generators for $\mathcal{AZ}_w$ but it is too large to be a basis of this vector space. For small weights, we find ad hoc arguments to produce a smaller set of generators but it does not work for arbitrary weights (see \S \ref{sec:without ACMPL}). Roughly speaking, in \cite[\S 2 and \S 3]{ND21} we have an algorithm which moves forward so that we can express any AMZV's as a linear combination of generators. But we lack some precise controls on coefficients in these expressions so that we cannot go backward and change bases. On the other hand, the transcendental part for AMZV's shares the same difficulties with the case of MZV's as noted before.

In this paper we use {\it a completely new approach} which is based on the study of alternating Carlitz multiple polylogarithms (ACMPL's for short) defined as follows. We put $\ell_0:=1$ and $\ell_d:=\prod_{i=1}^d (\theta-\theta^{q^i})$ for all $d \in \mathbb N$. For any tuple $\mathfrak{s}=(s_1,\ldots,s_r) \in \ensuremath \mathbb{N}^r$ and $\bm{\varepsilon}=(\varepsilon_1,\dots,\varepsilon_r) \in (\mathbb F_q^\times)^r$, we introduce the corresponding alternating Carlitz multiple polygarithm by
\begin{equation*} 
\Li\begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} := \sum\limits_{d_1> \dots > d_r\geq 0} \dfrac{\varepsilon_1^{d_1} \dots \varepsilon_r^{d_r} }{\ell_{d_1}^{s_1} \dots \ell_{d_r}^{s_r}}   \in K_{\infty}.
\end{equation*}

The key result is establishing a non trivial connection among AMZV's and ACMPL's which allows us to go back and forth among these objects (see Theorem \ref{thm:bridge}). To do so, following \cite[\S 2 and \S 3]{ND21} we use stuffle relations to develop an algebraic theory for ACMPL's and obtain a weak version of Brown's theorem, i.e., a set of generators for the $K$-vector space $\mathcal{AL}_w$ spanned by ACMPL's of weight $w$. We observe that this set of generators is exactly the same as that for AMZV's. Thus $\mathcal{AL}_w=\mathcal{AZ}_w$, which provides a dictionary among AMZV's and ACMPL's.

We then determine all $K$-linear relations among ACMPL's (see Theorem \ref{thm:ACMPL}). The proof we give here, although using similar tools as in \cite{ND21}, differs at some crucial points and requires {\it three new ingredients}.

The first new ingredient is constructing a suitable Hoffman-like basis $\mathcal{AS}_w$ of $\mathcal{AL}_w$. In fact, our transcendental method dictates that we have to find a full system of bases $\mathcal{AS}_w$ of $\mathcal{AL}_w$ indexed by weights $w$ with strong constraints as given in Theorem \ref{theorem: linear independence}. The failure of finding such a system of bases is the main obstruction for generalizing \cite[Theorem D]{ND21} (see \S \ref{sec: application AMZV} and \cite[Remark 6.3]{ND21} for more details).

The second new ingredient is formulating and proving (a strong version of) Brown's theorem for AMCPL's (see Theorem \ref{thm: strong Brown}). As mentioned before, the method in \cite{ND21} only yields a weak version of Brown's theorem for ACMPL's as the set of generators is not a basis. Roughly speaking, given any ACMPL's we can express it as a linear combination of generators. The fact that stuffle relations for ACMPL's are ``simpler" than shuffle relations for AMZV's give more precise information on coefficients of these expressions. Consequently, we show that a certain transition matrix is invertible and obtain Brown's theorem for ACMPL's. This settles the algebraic part for ACMPL's.

The last new ingredient is proving the transcendental part for ACMPL's in full generality, i.e., the ACMPL's in $\mathcal{AS}_w$ are linearly independent over $K$ (see Theorem \ref{thm: trans ACMPL}). We emphasize that we do need the full strength of the algebraic part to prove the transcendental part. The proof follows the same line in \cite[\S 4 and \S 5]{ND21} which is formulated in a more general setting in \S \ref{sec: dual motives}. First, we have to consider not only linear relations among ACMPL's in $\mathcal{AS}_w$ but also those among ACMPL's in $\mathcal{AS}_w$ and the suitable power $\widetilde \pi^w$ of the Carlitz period $\widetilde \pi$. Second, starting from such a non trivial relation we apply the Anderson-Brownawell-Papanikolas criterion in \cite{ABP04} and reduce to solve a system of $\sigma$-linear equations. While in \cite[\S 4 and \S 5]{ND21} this system does not have a non trivial solution which allows us to conclude, our system has a unique solution for even $w$ (i.e., $q-1$ divides $w$). This means that for such $w$ up to a scalar there is a unique linear relation among ACMPL's in $\mathcal{AS}_w$ and $\widetilde \pi^w$. The final step consists of showing that in this unique relation, the coefficient of $\widetilde \pi^w$ is non zero. Unexpectedly, this is a consequence of Brown's theorem for AMCPL's mentioned above.

\subsubsection{Plan of the paper}

We briefly explain the organization of the manuscript. 
\begin{itemize}
\item In \S \ref{sec:algebraic part} we recall the definition and basic properties of ACMPL's. We then develop an algebraic theory for these objects and obtain weak and strong Brown's theorems (see Proposition \ref{prop: weak Brown} and Theorem \ref{thm: strong Brown}).

\item In \S \ref{sec: dual motives} we generalize some transcendental results in \cite{ND21} and give statements in a more general setting (see Theorem \ref{theorem: linear independence}). 

\item In \S \ref{sec:transcendental part} we prove transcendental results for ACMPL's  and completely determine all linear relations among ACMPL's (see Theorems \ref{thm: trans ACMPL} and \ref{thm:ACMPL}). 

\item Finally, in \S \ref{sec:applications} we present two applications and prove the main results, i.e., Theorems \ref{thm: ZagierHoffman AMZV} and \ref{thm: ZagierHoffman}. The first application is to prove the aforementioned connection among ACMPL's and AMZV's and then to determine all linear relations among AMZV's in positive characteristic (see \S \ref{sec: application AMZV}). The second application is a proof of Zagier-Hoffman's conjectures in positive characteristic which generalizes the main results of \cite{ND21} (see \S \ref{sec: application MZV}). 
\end{itemize}


\subsection*{Acknowledgments}  
The first named author (B.-H. Im) was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No.~2020R1A2B5B01001835). Two of the authors (KN. L. and T. ND.) were partially supported by the Excellence Research Chair ``$L$-functions in positive characteristic and applications'' financed by the Normandy Region. The fourth author (T. ND.) was partially supported by the ANR Grant COLOSS ANR-19-CE40-0015-02. The fifth author (LH. P.) was supported the grant ICRTM.02-2021.05 funded by the International Center for Research and Postgraduate Training in Mathematics (VAST, Vietnam). 


\section{Weak and strong Brown's theorems for ACMPL's} \label{sec:algebraic part}

In this section we first extend the work of \cite{ND21} and develop an algebraic theory for ACMPL's. Next we prove a weak version of Brown's theorem for ACMPL's (see Theorem \ref{prop: weak Brown}) which gives a set of generators for the $K$-vector space spanned by ACMPL's of weight $w$. The techniques of Sections \ref{sec: power sums}-\ref{sec:weak Brown} are similar to \cite{ND21} and the expert reader could skip the details.

Contrary to what happens in \cite{ND21}, it turns out that the previous set of generators is too large to be a basis. Consequently, in \S \ref{sec:strong Brown} we introduce another set of generators and prove a strong version of Brown's theorem for ACMPL's (see Theorem \ref{thm: strong Brown}).

\subsection{Analogues of power sums} \label{sec: power sums}

\subsubsection{}

We recall and introduce some notation in \cite{ND21}. Letting $\mathfrak{s} = (s_1, \dotsc, s_n) \in \mathbb{N}^{n}$ and $\bm{\varepsilon} = (\varepsilon_1, \dotsc, \varepsilon_n) \in (\mathbb{F}_q^{\times})^{n}$, we set $\mathfrak{s}_- := (s_2, \dotsc, s_n)$ and $\bm{\varepsilon}_- := (\varepsilon_2, \dotsc, \varepsilon_n) $. A positive array $\begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} $ is an array of the form $$\begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  = \begin{pmatrix}
 \varepsilon_1 & \dotsb & \varepsilon_n \\
s_1 & \dotsb & s_n \end{pmatrix}.$$

For $i \in \mathbb{N}$ we define $T_i(\mathfrak{s})$ to be the tuple $(s_1+\dots+s_i,s_{i+1},\dots,s_n)$. Further, for tuples of positive integers $\mathfrak{s}, \mathfrak t$ and for $i \in \mathbb N$, if $T_i(\mathfrak{s}) \leq T_i(\mathfrak t)$, then $T_k(\mathfrak{s}) \leq T_k(\mathfrak t)$ for all $k \geq i$.
This notion extends straightforward to positive arrays.

Let $\mathfrak{s}=(s_1,\dots,s_r)$ be a tuple of positive integers. We denote by $0 \leq i \leq r$ the biggest integer such that $s_j \leq q$ for all $1 \leq j \leq i$ and define {\it the initial tuple} $\Init(\mathfrak{s})$ of $\mathfrak{s}$ to be the tuple 
	\[ \Init(\mathfrak{s}):=(s_1,\dots,s_i). \]
In particular, if $s_1>q$, then $i=0$ and $\Init(\mathfrak{s})$ is the empty tuple.

For two different tuples $\mathfrak{s}$ and $\mathfrak t$, we consider the lexicographical order for initial tuples and write $\Init(\mathfrak t) \preceq \Init(\mathfrak{s})$ (resp. $\Init(\mathfrak t) \prec \Init(\mathfrak{s})$, $\Init(\mathfrak t) \succeq \Init(\mathfrak{s})$ and $\Init(\mathfrak t) \succ \Init(\mathfrak{s})$). 

\subsubsection{} We recall the power sums studied by Thakur \cite{Tha10}. For $d \in \mathbb{Z}$ and for $\mathfrak{s}=(s_1,\dots,s_n) \in \ensuremath \mathbb{N}^n$ we introduce
\begin{equation*}
    S_d(\mathfrak{s}) =  \sum\limits_{\substack{a_1, \dots, a_n \in A_{+} \\ d = \deg a_1> \dots > \deg a_n\geq 0}} \dfrac{1}{a_1^{s_1} \dots a_n^{s_n}} \in K
\end{equation*}
and 
\begin{equation*}
    S_{<d}(\mathfrak s) = \sum\limits_{\substack{a_1, \dots, a_n \in A_{+} \\ d > \deg a_1> \dots > \deg a_n\geq 0}} \dfrac{1}{a_1^{s_1} \dots a_n^{s_n}} \in K.
\end{equation*}


We also recall $\ell_0 := 1$ and $\ell_d := \prod^d_{i=1}(\theta - \theta^{q^i})$ for all $d \in \mathbb{N}$. Letting $\mathfrak s = (s_1 , \dots, s_n) \in \mathbb{N}^n$, for $d \in \mathbb{Z}$, we define analogues of power sums by
\begin{equation*}
        \Si_d(\mathfrak s) = \sum\limits_{d=d_1> \dots > d_n\geq 0} \dfrac{1}{\ell_{d_1}^{s_1} \dots \ell_{d_n}^{s_n}} \in K,
\end{equation*}
and 
\begin{equation*}
        \Si_{<d}(\mathfrak s) = \sum\limits_{d>d_1> \dots > d_n\geq 0} \dfrac{1 }{\ell_{d_1}^{s_1} \dots \ell_{d_n}^{s_n}} \in K.
\end{equation*}

Let $\begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  =  \begin{pmatrix}
 \varepsilon_1 & \dots & \varepsilon_n \\
s_1 & \dots & s_n \end{pmatrix} $ be a positive array. For $d \in \mathbb{Z}$, we define 
\begin{equation*}
S_d \begin{pmatrix}
\bm{\varepsilon} \\ \mathfrak{s}
\end{pmatrix}  = \sum\limits_{\substack{a_1, \dots, a_n \in A_{+} \\ d = \deg a_1> \dots > \deg a_n\geq 0}} \dfrac{\varepsilon_1^{\deg a_1} \dots \varepsilon_n^{\deg a_n }}{a_1^{s_1} \dots a_n^{s_n}} \in K
\end{equation*}
and 
\begin{equation*}
    S_{<d} \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  = \sum\limits_{\substack{a_1, \dots, a_n \in A_{+} \\ d > \deg a_1> \dots > \deg a_n\geq 0}} \dfrac{\varepsilon_1^{\deg a_1} \dots \varepsilon_n^{\deg a_n }}{a_1^{s_1} \dots a_n^{s_n}} \in K.
\end{equation*}
We also introduce
\begin{equation*}
        \Si_d \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  = \sum\limits_{d=d_1> \dots > d_n\geq 0} \dfrac{\varepsilon_1^{d_1} \dots \varepsilon_n^{d_n} }{\ell_{d_1}^{s_1} \dots \ell_{d_n}^{s_n}} \in K,
\end{equation*}
and 
\begin{equation*}
        \Si_{<d} \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  = \sum\limits_{d>d_1> \dots > d_n\geq 0} \dfrac{\varepsilon_1^{d_1} \dots \varepsilon_n^{d_n} }{\ell_{d_1}^{s_1} \dots \ell_{d_n}^{s_n}} \in K.
\end{equation*}
One verifies easily the following formulas:
\begin{align*}
\Si_{d} 
    \begin{pmatrix} \varepsilon \\ s  \end{pmatrix}  
&= \varepsilon^d \Si_d(s),\\
\Si_d \begin{pmatrix}
 1& \dots & 1 \\
s_1 & \dots & s_n \end{pmatrix}  &= \Si_{d}(s_1, \dots, s_n),\\
\Si_{<d} \begin{pmatrix}
 1& \dots & 1 \\
s_1 & \dots & s_n \end{pmatrix}  &= \Si_{<d}(s_1, \dots, s_n),\\
\Si_{d} \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  &= \Si_{d} \begin{pmatrix}
 \varepsilon_1  \\
s_1  \end{pmatrix} \Si_{<d} \begin{pmatrix}
 \bm{\varepsilon}_{-}  \\
\mathfrak{s}_{-}  \end{pmatrix} .
\end{align*}
Then we define the alternating Carlitz multiple polygarithm (ACMPL for short) as follows
\begin{equation*}
    \Li \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  = \sum \limits_{d \geq 0} \Si_d \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  = \sum\limits_{d_1> \dots > d_n\geq 0} \dfrac{\varepsilon_1^{d_1} \dots \varepsilon_n^{d_n} }{\ell_{d_1}^{s_1} \dots \ell_{d_n}^{s_n}}   \in K_{\infty}.
\end{equation*}
We agree also that
$\Li \begin{pmatrix}
 \emptyset  \\
 \emptyset  \end{pmatrix}  = 1$. We call $\depth(\mathfrak{s}) = n$ the depth, $w(\mathfrak{s}) = s_1 + \dots + s_n$ the weight and $\chi(\bm{\varepsilon}) = \varepsilon_1  \dots  \varepsilon_n$ the character of  $\Li \begin{pmatrix}
 \bm{\varepsilon}  \\ \mathfrak{s}  \end{pmatrix}$.
 
\begin{lemma} \label{agree}
For all $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}$ as above such that $s_i \leq q$ for all $i$, we have 
\begin{equation*}
   S_d \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  =  \Si_d \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  \quad \text{for all } d \in \mathbb{Z}.
\end{equation*}
Therefore, 
\begin{equation*}
    \zeta_A  \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  = \Li \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} .
\end{equation*}
\end{lemma}
\begin{proof}
We denote by $\mathcal{J}$ the set of all positive arrays $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  =  \begin{pmatrix}
 \varepsilon_1 & \dots & \varepsilon_n \\
s_1 & \dots & s_n \end{pmatrix} $ for some $n$ such that $s_1, \dots, s_n \leq q$.

The second statement follows at once from the first statement. We prove the first statement by induction on $\depth(\mathfrak{s})$. For $\depth(\mathfrak{s}) = 1$, we let $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  =  \begin{pmatrix}
 \varepsilon \\
s  \end{pmatrix} $ with $s \le q$. It follows from  special cases of power sums in \cite[\S 3.3]{Tha09} that for all $d \in \mathbb{Z}$,
\begin{equation*}
    S_{d} \begin{pmatrix}
 \varepsilon \\
s  \end{pmatrix}  =  \dfrac{\varepsilon^d}{\ell^s_d} = \Si_{d} \begin{pmatrix}
 \varepsilon \\
s  \end{pmatrix} .
\end{equation*} Suppose that the first statement holds for all arrays $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  \in \mathcal{J}$ with $\depth(\mathfrak{s}) = n - 1$ and for all $ d \in \mathbb{Z}$. Let $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  =  \begin{pmatrix}
 \varepsilon_1 & \dots & \varepsilon_n \\
s_1 & \dots & s_n \end{pmatrix} $ be an element of $\mathcal{J}$. Note that if $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  \in \mathcal{J}$, then $ \begin{pmatrix}
 \bm{\varepsilon}_{-}  \\
\mathfrak{s}_{-}  \end{pmatrix}  \in \mathcal{J}$. It follows from induction hypothesis and the fact $s_1 \leq q$ that for all $d \in \mathbb{Z}$ 
\begin{equation*}
    S_d \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  = S_d \begin{pmatrix}
 \varepsilon_1  \\
s_1  \end{pmatrix}  S_{<d} \begin{pmatrix}
 \bm{\varepsilon}_{-}  \\
\mathfrak{s}_{-}  \end{pmatrix} 
    = \Si_d \begin{pmatrix}
 \varepsilon_1  \\
s_1  \end{pmatrix}  \Si_{<d} \begin{pmatrix}
 \bm{\varepsilon}_{-}  \\
\mathfrak{s}_{-}  \end{pmatrix}  = \Si_d \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} .
\end{equation*}
This proves the lemma.
\end{proof}

Let $\begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}$, $\begin{pmatrix}
 \bm{\epsilon}  \\
\mathfrak{t}  \end{pmatrix}$ be two positive arrays. We set $s_i = 0$ and $\varepsilon_i = 1$ for all $i > \depth(\mathfrak{s})$; $t_i = 0$ and $\epsilon_i = 1$ for all $i > \depth(\mathfrak{t})$. We define the following operation
\begin{equation*}
    \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} + \begin{pmatrix}
 \bm{\epsilon}  \\
\mathfrak{t}  \end{pmatrix} := \begin{pmatrix}
 \bm{\varepsilon} \bm{\epsilon}  \\
\mathfrak{s} + \mathfrak{t}  \end{pmatrix},
\end{equation*}
where $\bm{\varepsilon} \bm{\epsilon}$ and $\mathfrak{s} + \mathfrak{t}$ are defined by component multiplication and component addition, respectively. We say that $\begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} \leq \begin{pmatrix}
 \bm{\epsilon}  \\
\mathfrak{t}  \end{pmatrix} $
 if the following conditions are satisfied:
\begin{enumerate}
    \item $\chi(\bm{\varepsilon}) = \chi(\bm{\epsilon})$,
    \item $w(\mathfrak{s}) = w(\mathfrak{t})$,
    \item $s_1 + \dotsb + s_i \leq t_1 + \dotsb + t_i$ for all $i \in \ensuremath \mathbb{N}$.
\end{enumerate}

\subsubsection{}

We now consider some formulas related to analogues of power sums. It is easily seen that 
\begin{equation*}
  \Si_d \begin{pmatrix}
 \varepsilon \\
s  \end{pmatrix}   \Si_d \begin{pmatrix}
 \epsilon  \\
t  \end{pmatrix}   = \Si_d \begin{pmatrix}
 \varepsilon \epsilon \\
s + t  \end{pmatrix} ,   
\end{equation*}
hence, for  $\mathfrak{t} = (t_1, \dots, t_n)$, 
\begin{equation}\label{redsum}
  \Si_d \begin{pmatrix}
 \varepsilon \\
s  \end{pmatrix}   \Si_d \begin{pmatrix}
 \bm{\epsilon}  \\
\mathfrak{t}  \end{pmatrix}   = \Si_d \begin{pmatrix}
 \varepsilon \epsilon_1 & \bm{\epsilon}_{-}  \\
s + t_1  & \mathfrak{t}_{-} \end{pmatrix}. 
\end{equation}
More generally, we deduce the following proposition which will be used frequently later.
\begin{proposition} \label{polysums}
Let $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} $, $ \begin{pmatrix}
 \bm{\epsilon}  \\
\mathfrak{t}  \end{pmatrix} $ be two positive arrays. Then 
\begin{enumerate}
    \item There exist $f_i \in \mathbb{F}_p$ and arrays $ \begin{pmatrix}
 \bm{\mu}_i  \\
\mathfrak{u}_i  \end{pmatrix} $ with $ \begin{pmatrix}
 \bm{\mu}_i  \\
\mathfrak{u}_i  \end{pmatrix}  \leq  \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  +  \begin{pmatrix}
 \bm{\epsilon}  \\
\mathfrak{t}  \end{pmatrix}  $ and $\depth(\mathfrak{u}_i) \leq \depth(\mathfrak{s}) + \depth(\mathfrak{t})$ for all $i$  such that 
    \begin{equation*}
        \Si_d \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} \Si_d \begin{pmatrix}
 \bm{\epsilon}  \\
\mathfrak{t}  \end{pmatrix}  = \sum \limits_i f_i \Si_d \begin{pmatrix}
 \bm{\mu}_i  \\
\mathfrak{u}_i  \end{pmatrix}  \quad \text{for all } d \in \mathbb{Z}.
    \end{equation*}
    \item There exist $f'_i \in \mathbb{F}_p$ and arrays $ \begin{pmatrix}
 \bm{\mu}'_i  \\
\mathfrak{u}'_i  \end{pmatrix} $ with $ \begin{pmatrix}
 \bm{\mu}'_i  \\
\mathfrak{u}'_i  \end{pmatrix}  \leq  \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  +  \begin{pmatrix}
 \bm{\epsilon}  \\
\mathfrak{t}  \end{pmatrix}  $ and $\depth(\mathfrak{u}'_i) \leq \depth(\mathfrak{s}) + \depth(\mathfrak{t})$ for all $i$  such that 
    \begin{equation*}
        \Si_{<d} \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} \Si_{<d} \begin{pmatrix}
 \bm{\epsilon}  \\
\mathfrak{t}  \end{pmatrix}  = \sum \limits_i f'_i \Si_{<d} \begin{pmatrix}
 \bm{\mu}'_i  \\
\mathfrak{u}'_i  \end{pmatrix}  \quad \text{for all } d \in \mathbb{Z}.
    \end{equation*}
        \item There exist $f''_i \in \mathbb{F}_p$ and arrays $ \begin{pmatrix}
 \bm{\mu}''_i  \\
\mathfrak{u}''_i  \end{pmatrix} $ with $ \begin{pmatrix}
 \bm{\mu}''_i  \\
\mathfrak{u}''_i  \end{pmatrix}  \leq  \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  +  \begin{pmatrix}
 \bm{\epsilon}  \\
\mathfrak{t}  \end{pmatrix}  $ and $\depth(\mathfrak{u}''_i) \leq \depth(\mathfrak{s}) + \depth(\mathfrak{t})$ for all $i$  such that 
    \begin{equation*}
        \Si_d \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} \Si_{<d} \begin{pmatrix}
 \bm{\epsilon}  \\
\mathfrak{t}  \end{pmatrix}  = \sum \limits_i f''_i \Si_d \begin{pmatrix}
 \bm{\mu}''_i  \\
\mathfrak{u}''_i  \end{pmatrix}  \quad \text{for all } d \in \mathbb{Z}.
    \end{equation*}
\end{enumerate}
\end{proposition}
\begin{proof}
The proof follows the same line as in \cite[Proposition 2.1]{ND21}. We omit the details.
\end{proof}


We denote by $\mathcal{AL}$ the $K$-vector space generated by the ACMPL's and by $\mathcal{AL}_w$ the $K$-vector space generated by the ACMPL's of weight $w$. It follows from Proposition~\ref{polysums} that $\mathcal{AL}$ is a $K$-algebra under the multiplication of $K_{\infty}$.


\subsection{Operators $\mathcal B^*$, $\mathcal C$ and $\mathcal{BC}$} \label{sec:Todd}

In this section we define operators $\mathcal B^*$ and $\mathcal C$ of Todd \cite{Tod18} and the operator $\mathcal{BC}$ of Ngo Dac  \cite{ND21} in the case of ACMPL's.

\begin{definition}
A binary relation is a $K$-linear combination of the form 
\begin{equation*}
    \sum \limits_i a_i \Si_d \begin{pmatrix}
 \bm{\varepsilon}_i  \\
\mathfrak{s}_i  \end{pmatrix}  + \sum \limits_i b_i \Si_{d+1} \begin{pmatrix}
 \bm{\epsilon}_i  \\
\mathfrak{t}_i  \end{pmatrix}  =0 \quad \text{for all } d \in \mathbb{Z},
\end{equation*}
where $a_i,b_i \in K$ and $ \begin{pmatrix}
 \bm{\varepsilon}_i  \\
\mathfrak{s}_i  \end{pmatrix} ,  \begin{pmatrix}
 \bm{\epsilon}_i  \\
\mathfrak{t}_i  \end{pmatrix} $ are positive arrays of the same weight. 

A binary relation is called a fixed relation if $b_i = 0$ for all $i$.
\end{definition}

We denote by $\mathfrak{BR}_{w}$ the set of all binary relations of weight $w$. One verifies at once that $\mathfrak{BR}_{w}$ is a $K$-vector space under the addition and multiplication of $K$. It follows from the fundamental relation in \cite[\S 3.4.6]{Tha09} and Lemma \ref{agree}, an important example of binary relations
\begin{equation*}
 R_{\varepsilon} \colon \quad  \Si_d \begin{pmatrix}
 \varepsilon\\
q  \end{pmatrix}  + \varepsilon^{-1}D_1 \Si_{d+1} \begin{pmatrix}
 \varepsilon& 1 \\
1 & q-1  \end{pmatrix}  =0, 
\end{equation*}
where $D_1 = \theta^q - \theta$.

For later definitions, let $R \in \mathfrak{BR}_w$ be a binary relation of the form 
\begin{equation*}
    R(d) \colon \quad \sum \limits_i a_i \Si_d \begin{pmatrix}
 \bm{\varepsilon}_i  \\
\mathfrak{s}_i  \end{pmatrix}  + \sum \limits_i b_i \Si_{d+1} \begin{pmatrix}
 \bm{\epsilon}_i  \\
\mathfrak{t}_i  \end{pmatrix}  =0,
\end{equation*}
where $a_i,b_i \in K$ and $ \begin{pmatrix}
 \bm{\varepsilon}_i  \\
\mathfrak{s}_i  \end{pmatrix} ,  \begin{pmatrix}
 \bm{\epsilon}_i  \\
\mathfrak{t}_i  \end{pmatrix} $ are positive arrays of the same weight. We now define some operators on $K$-vector spaces of binary relations.

\subsubsection{Operators $\mathcal B^*$}

Let $ \begin{pmatrix}
 \sigma  \\
v  \end{pmatrix} $ be a positive array. We define an operator
\begin{equation*}
    \mathcal B^*_{\sigma,v} \colon \mathfrak{BR}_{w} \longrightarrow \mathfrak{BR}_{w+v}
\end{equation*}
as follows: for each $R \in \mathfrak{BR}_{w}$,
the image $\mathcal B^*_{\sigma,v}(R) = \Si_d \begin{pmatrix}
 \sigma  \\
v  \end{pmatrix} \sum_{j < d} R(j)$ is a fixed relation of the form 
\begin{align*}
    0 &= \Si_d \begin{pmatrix}
 \sigma  \\
v  \end{pmatrix}   \left(\sum \limits_ia_i \Si_{<d} \begin{pmatrix}
 \bm{\varepsilon}_i  \\
\mathfrak{s}_i  \end{pmatrix}  + \sum \limits_i  b_i \Si_{<d+1} \begin{pmatrix}
 \bm{\epsilon}_i  \\
\mathfrak{t}_i  \end{pmatrix} \right)  \\
    &= \sum \limits_i a_i \Si_d \begin{pmatrix}
 \sigma  \\
v  \end{pmatrix} \Si_{<d} \begin{pmatrix}
 \bm{\varepsilon}_i  \\
\mathfrak{s}_i  \end{pmatrix}  + \sum \limits_i  b_i \Si_d \begin{pmatrix}
 \sigma  \\
v  \end{pmatrix}  \Si_{<d} \begin{pmatrix}
 \bm{\epsilon}_i  \\
\mathfrak{t}_i  \end{pmatrix}  + \sum \limits_i  b_i \Si_d \begin{pmatrix}
 \sigma  \\
v  \end{pmatrix}  \Si_{d} \begin{pmatrix}
 \bm{\epsilon}_i  \\
\mathfrak{t}_i  \end{pmatrix} \\
    &= \sum \limits_i a_i \Si_d \begin{pmatrix}
\sigma & \bm{\varepsilon}_i  \\
v& \mathfrak{s}_i  \end{pmatrix}  + \sum \limits_i  b_i \Si_d \begin{pmatrix}
\sigma & \bm{\epsilon}_i  \\
v& \mathfrak{t}_i  \end{pmatrix}  + \sum \limits_i  b_i  \Si_d \begin{pmatrix}
 \sigma \epsilon_{i1}  & \bm{\epsilon}_{i-} \\
v + t_{i1} & \mathfrak{t}_{i-} \end{pmatrix} .
\end{align*}
The last equality follows from \eqref{redsum}.

Let $ \begin{pmatrix}
 \Sigma  \\
V  \end{pmatrix}  =  \begin{pmatrix}
 \sigma_1 & \dots & \sigma_n \\
v_1 & \dots & v_n \end{pmatrix} $ be a positive array. We define an operator $\mathcal{B}^*_{\Sigma,V}(R) $ by
\begin{equation*}
    \mathcal B^*_{\Sigma,V}(R) = \mathcal B^*_{\sigma_1,v_1} \circ \dots \circ \mathcal B^*_{\sigma_n,v_n}(R).
\end{equation*}
\


\begin{lemma} \label{polybesao}
Let $ \begin{pmatrix}
 \Sigma  \\
V  \end{pmatrix}  =  \begin{pmatrix}
 \sigma_1 & \dots & \sigma_n \\
v_1 & \dots & v_n \end{pmatrix} $ be a positive array. Then $\mathcal B^*_{\Sigma,V}(R)$ is of the form
\begin{align*}
 \sum \limits_i a_i \Si_d \begin{pmatrix}
\Sigma & \bm{\varepsilon}_i  \\
V& \mathfrak{s}_i  \end{pmatrix} & + \sum \limits_i  b_i \Si_d \begin{pmatrix}
\Sigma & \bm{\epsilon}_i  \\
V& \mathfrak{t}_i  \end{pmatrix}  
\\ & + \sum \limits_i  b_i \Si_d  \begin{pmatrix}
 \sigma_1 & \dots & \sigma_{n-1} & \sigma_n \epsilon_{i1} &\bm{\epsilon}_{i-}  \\
v_1 & \dots & v_{n-1} & v_n+ t_{i1} & \mathfrak{t}_{i-}  \end{pmatrix}  = 0 .
\end{align*}
\end{lemma}

\begin{proof}
The proof is straightforward. We omit this proof.
\end{proof}

\subsubsection{Operators $\mathcal C$}

Let $ \begin{pmatrix}
 \Sigma  \\
V  \end{pmatrix} $ be a positive array of weight $v$. We define an operator
\begin{equation*}
     \mathcal C_{\Sigma,V}(R) \colon \mathfrak{BR}_{w} \longrightarrow \mathfrak{BR}_{w+v}
\end{equation*}
as follows: for each $R \in \mathfrak{BR}_{w}$,
the image $\mathcal C_{\Sigma,V}(R) = R(d) \Si_{<d+1} \begin{pmatrix}
 \Sigma  \\
V  \end{pmatrix} $ is a binary relation of the form 
\begin{align*}
    0 &= \left( \sum \limits_i a_i \Si_d \begin{pmatrix}
 \bm{\varepsilon}_i  \\
\mathfrak{s}_i  \end{pmatrix}  + \sum \limits_i b_i \Si_{d+1} \begin{pmatrix}
 \bm{\epsilon}_i  \\
\mathfrak{t}_i  \end{pmatrix} \right) \Si_{<d+1} \begin{pmatrix}
 \Sigma  \\
V  \end{pmatrix}   \\
    &= \sum \limits_i a_i \Si_d \begin{pmatrix}
 \bm{\varepsilon}_i  \\
\mathfrak{s}_i  \end{pmatrix} \Si_{d} \begin{pmatrix}
 \Sigma  \\
V  \end{pmatrix}  + \sum \limits_i a_i \Si_d \begin{pmatrix}
 \bm{\varepsilon}_i  \\
\mathfrak{s}_i  \end{pmatrix} \Si_{<d} \begin{pmatrix}
 \Sigma  \\
V  \end{pmatrix}  + \sum \limits_i b_i \Si_{d+1} \begin{pmatrix}
 \bm{\epsilon}_i  \\
\mathfrak{t}_i  \end{pmatrix} \Si_{<d+1} \begin{pmatrix}
 \Sigma  \\
V  \end{pmatrix} \\
    &= \sum \limits_i f_i \Si_d \begin{pmatrix}
 \bm{\mu}_i  \\
\mathfrak{u}_i  \end{pmatrix}  + \sum \limits_i f'_i \Si_{d+1} \begin{pmatrix}
 \bm{\mu}'_i  \\
\mathfrak{u}'_i  \end{pmatrix} .
\end{align*}
The last equality follows from Proposition \ref{polysums}.


In particular, the following proposition gives the form of $\mathcal C_{\Sigma,V}(R_{\varepsilon})$.

\begin{proposition} \label{polycer1}
Let $ \begin{pmatrix}
 \Sigma  \\ V  \end{pmatrix}$ be a positive array with $V = (v_1,V_{-})$ and $\Sigma = (\sigma_1, \Sigma_{-})$. Then $\mathcal C_{\Sigma,V}(R_{\varepsilon})$ is of the form 
\begin{equation*}
  \Si_d \begin{pmatrix}
 \varepsilon\sigma_1 & \Sigma_{-} \\
q + v_1 & V_{-} \end{pmatrix}  +  \Si_d \begin{pmatrix}
 \varepsilon& \Sigma \\
q & V \end{pmatrix}  + \sum \limits_i b_i \Si_{d+1} \begin{pmatrix}
\varepsilon& \bm{\epsilon}_i  \\
1 & \mathfrak{t}_i  \end{pmatrix}  =0,
\end{equation*}
where $b_i \in K$ and $ \begin{pmatrix}
 \bm{\epsilon}_i  \\
\mathfrak{t}_i  \end{pmatrix} $ are positive arrays satisfying $ \begin{pmatrix}
 \bm{\epsilon}_i  \\
\mathfrak{t}_i  \end{pmatrix}  \leq  \begin{pmatrix}
 1  \\
q - 1 \end{pmatrix}  +  \begin{pmatrix}
 \Sigma  \\
V  \end{pmatrix} $ for all $i$.
\end{proposition}

\begin{proof}
We see that $\mathcal C_{\Sigma,V}(R_{\varepsilon})$ is of the form
\begin{equation*}
    \Si_d \begin{pmatrix}
\varepsilon \\
q  \end{pmatrix} \Si_{d} \begin{pmatrix}
 \Sigma  \\
V  \end{pmatrix}  + \Si_d \begin{pmatrix}
 \varepsilon \\
q  \end{pmatrix} \Si_{<d} \begin{pmatrix}
 \Sigma  \\
V  \end{pmatrix}  + \varepsilon^{-1} D_1 \Si_{d+1} \begin{pmatrix}
\varepsilon&  1  \\
1 & q-1 \end{pmatrix} \Si_{<d+1} \begin{pmatrix}
 \Sigma  \\
V  \end{pmatrix}  = 0.
\end{equation*}
It follows from \eqref{redsum} and Proposition \ref{polysums} that 
\begin{align*}
        \Si_d \begin{pmatrix}
\varepsilon \\
q  \end{pmatrix} \Si_{d} \begin{pmatrix}
 \Sigma  \\
V  \end{pmatrix}  + \Si_d \begin{pmatrix}
 \varepsilon \\
q  \end{pmatrix} \Si_{<d} \begin{pmatrix}
 \Sigma  \\
V  \end{pmatrix}  &=   \Si_d \begin{pmatrix}
 \varepsilon\sigma_1 & \Sigma_{-} \\
q + v_1 & V_{-} \end{pmatrix}  +  \Si_d \begin{pmatrix}
 \varepsilon& \Sigma \\
q & V \end{pmatrix},  \\
  \varepsilon^{-1}D_1 \Si_{d+1} \begin{pmatrix}
\varepsilon&  1  \\
1 & q-1 \end{pmatrix} \Si_{<d+1} \begin{pmatrix}
 \Sigma  \\
V  \end{pmatrix}   &= \sum \limits_i b_i \Si_{d+1} \begin{pmatrix}
 \varepsilon& \bm{\epsilon}_i  \\
1 & \mathfrak{t}_i  \end{pmatrix} ,
\end{align*}
where $b_i \in K$ and $ \begin{pmatrix}
 \bm{\epsilon}_i  \\
\mathfrak{t}_i  \end{pmatrix} $ are positive arrays satisfying $ \begin{pmatrix}
 \bm{\epsilon}_i  \\
\mathfrak{t}_i  \end{pmatrix}  \leq  \begin{pmatrix}
 1  \\
q - 1 \end{pmatrix}  +  \begin{pmatrix}
 \Sigma  \\
V  \end{pmatrix} $ for all $i$.
This proves the proposition.
\end{proof}

\subsubsection{Operators $\mathcal{BC}$}

Let $\varepsilon \in \mathbb{F}_q^{\times}$. We define an operator
\begin{equation*}
   \mathcal{BC}_{\varepsilon,q} \colon \mathfrak{BR}_{w} \longrightarrow \mathfrak{BR}_{w+q}
\end{equation*}
as follows: for each $R \in \mathfrak{BR}_{w}$,
the image $\mathcal{BC}_{\varepsilon,q}(R)$ is a binary relation given by
\begin{align*}
    \mathcal{BC}_{\varepsilon,q}(R) = \mathcal B^*_{\varepsilon,q}(R) - \sum\limits_i b_i \mathcal C_{\bm{\epsilon}_i,\mathfrak{t}_i} (R_{\varepsilon}).
\end{align*}

Let us clarify the definition of $\mathcal{BC}_{\varepsilon,q}$. We know that $\mathcal B^*_{\varepsilon,q}(R)$ is of the form 
\begin{equation*}
    \sum \limits_i a_i \Si_d \begin{pmatrix}
\varepsilon& \bm{\varepsilon}_i  \\
q& \mathfrak{s}_i  \end{pmatrix}  + \sum \limits_i  b_i \Si_d \begin{pmatrix}
\varepsilon& \bm{\epsilon}_i  \\
q& \mathfrak{t}_i  \end{pmatrix}  + \sum \limits_i  b_i \Si_d \begin{pmatrix}
 \varepsilon\epsilon_{i1} &  \bm{\epsilon}_{i-} \\
q + t_{i1} & \mathfrak{t}_{i-} \end{pmatrix} = 0.
\end{equation*}
Moreover, $\mathcal C_{\bm{\epsilon}_i,\mathfrak{t}_i} (R_{\varepsilon})$ is of the form
\begin{equation*}
Si_d \begin{pmatrix}
\varepsilon& \bm{\epsilon}_i  \\
q& \mathfrak{t}_i  \end{pmatrix}  +  \Si_d \begin{pmatrix}
 \varepsilon\epsilon_{i1} &  \bm{\epsilon}_{i-} \\
q + t_{i1} & \mathfrak{t}_{i-} \end{pmatrix}  + \varepsilon^{-1}D_1 \Si_{d+1} \begin{pmatrix}
 \varepsilon \\
1  \end{pmatrix} \Si_{<d+1} \begin{pmatrix}
1  \\
q-1  \end{pmatrix} \Si_{<d+1} \begin{pmatrix}
\bm{\epsilon}_i  \\
\mathfrak{t}_i  \end{pmatrix} = 0.
\end{equation*}
Combining with Proposition \ref{polysums} (2), we have that $\mathcal{BC}_{\varepsilon,q}(R)$ is of the form
\begin{equation*}
   \sum \limits_i a_i \Si_d \begin{pmatrix}
\varepsilon& \bm{\varepsilon}_i  \\
q& \mathfrak{s}_i  \end{pmatrix}  + \sum \limits_{i,j} b_{ij} \Si_{d+1} \begin{pmatrix}
\varepsilon& \bm{\epsilon}_{ij}  \\
1& \mathfrak{t}_{ij}  \end{pmatrix} =0,
\end{equation*}
where $b_{ij} \in K$ and $ \begin{pmatrix}
\bm{\epsilon}_{ij}  \\
\mathfrak{t}_{ij}  \end{pmatrix} $ are positive arrays satisfying $ \begin{pmatrix}
\bm{\epsilon}_{ij}  \\
\mathfrak{t}_{ij}  \end{pmatrix}  \leq  \begin{pmatrix}
1  \\
q-1  \end{pmatrix}  +  \begin{pmatrix}
\bm{\epsilon}_{i}  \\
\mathfrak{t}_{i}  \end{pmatrix} $ for all $j$.

\subsection{A weak version of Brown's theorem for ACMP's} \label{sec:weak Brown}

\subsubsection{Preparatory results}

\begin{proposition}\label{polydecom}
1) Let $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  =  \begin{pmatrix}
 \varepsilon_1 & \dots & \varepsilon_n \\
s_1 & \dots & s_n \end{pmatrix} $ be a positive array such that $\Init(\mathfrak{s}) = (s_1, \dots, s_{k-1})$ for some $1 \leq k \leq n$, and let $\varepsilon$ be an element in $\mathbb{F}_q^{\times}$.  Then $\Li \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} $ can be decomposed as follows
    \begin{equation*}
    \Li \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} 
    = \underbrace{ - \Li \begin{pmatrix}
 \bm{\varepsilon}'  \\
\mathfrak{s}'  \end{pmatrix} }_\text{type 1} + \underbrace{\sum\limits_i b_i\Li \begin{pmatrix}
 \bm{\epsilon}_i'  \\
\mathfrak{t}'_i  \end{pmatrix} }_\text{type 2} + \underbrace{\sum\limits_i c_i\Li \begin{pmatrix}
 \bm{\mu}_i  \\
\mathfrak{u}_i  \end{pmatrix} }_\text{type 3}  ,
    \end{equation*}
    where $ b_i, c_i \in A$ divisible by $D_1$ such that for all $i$, the following properties are satisfied:
    \begin{itemize}
        \item For all arrays $ \begin{pmatrix}
 \bm{\epsilon}  \\
\mathfrak{t}  \end{pmatrix} $ appearing on the right hand side,
        \begin{equation*}
            \depth(\mathfrak{t}) \geq \depth(\mathfrak{s}) \quad \text{and} \quad T_k(\mathfrak{t}) \leq T_k(\mathfrak{s}).
     \end{equation*} 
     \item For the array $ \begin{pmatrix}
 \bm{\varepsilon}'  \\
\mathfrak{s}'  \end{pmatrix} $ of type $1$ with respect to $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} $, we have
\begin{align*}
\begin{pmatrix}
 \bm{\varepsilon}'  \\
\mathfrak{s}'  \end{pmatrix} =
\begin{pmatrix}
\varepsilon_1 & \dots & \varepsilon_{k-1} & \varepsilon & \varepsilon^{-1}\varepsilon_{k} & \varepsilon_{k+1} & \dots & \varepsilon_n \\
s_1 & \dots & s_{k-1} & q &  s_k- q & s_{k+1} & \dots & s_n \end{pmatrix}. 
\end{align*}
Moreover, for all $k \leq \ell \leq n$,
        \begin{equation*}
s'_{1} +  \dots + s'_\ell < s_1 +  \dots + s_\ell.
     \end{equation*}
     
     
\item For the array $ \begin{pmatrix}
 \bm{\epsilon}'  \\
\mathfrak{t}'  \end{pmatrix} $ of type $2$ with respect to $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} $, for all $k \leq \ell \leq n$,
        \begin{equation*}
t'_{1} +  \dots + t'_\ell < s_1 +  \dots + s_\ell.
     \end{equation*}
        
        \item For the array $ \begin{pmatrix}
 \bm{\mu}  \\
\mathfrak{u}  \end{pmatrix} $ of type $3$ with respect to $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} $, we have $\Init(\mathfrak{s}) \prec\Init(\mathfrak{u})$.
    \end{itemize}
    
\noindent 2) Let $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  =  \begin{pmatrix}
 \varepsilon_1 & \dots & \varepsilon_k \\
s_1 & \dots & s_k \end{pmatrix} $ be a positive array such that $\Init(\mathfrak{s}) = \mathfrak{s}$ and $s_k = q$. Then $\Li \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} $ can be decomposed as follows:
    \begin{equation*}
    \Li \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} 
    =   \underbrace{\sum\limits_i b_i\Li \begin{pmatrix}
 \bm{\epsilon}'_i  \\
\mathfrak{t}'_i  \end{pmatrix} }_\text{type 2} + \underbrace{\sum\limits_i c_i\Li \begin{pmatrix}
 \bm{\mu}_i  \\
\mathfrak{u}_i  \end{pmatrix} }_\text{type 3}  ,
    \end{equation*}
    where $ b_i, c_i \in A$ divisible by $D_1$ such that for all $i$, the following properties are satisfied:
    \begin{itemize}
        \item For all arrays $ \begin{pmatrix}
 \bm{\epsilon}  \\
\mathfrak{t}  \end{pmatrix} $ appearing on the right hand side,
        \begin{equation*}
            \depth(\mathfrak{t}) \geq \depth(\mathfrak{s}) \quad \text{and} \quad T_k(\mathfrak{t}) \leq T_k(\mathfrak{s}).
     \end{equation*} 
    
     
     
\item For the array $ \begin{pmatrix}
 \bm{\epsilon}'  \\
\mathfrak{t}'  \end{pmatrix} $ of type $2$ with respect to $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} $, 
        \begin{equation*}
t'_{1} +  \dots + t'_k < s_1 +  \dots + s_k.
     \end{equation*}
        
        \item For the array $ \begin{pmatrix}
 \bm{\mu}  \\
\mathfrak{u}  \end{pmatrix} $ of type $3$ with respect to $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} $, we have $\Init(\mathfrak{s}) \prec\Init(\mathfrak{u})$.
\end{itemize}
\end{proposition}


\begin{proof}
For Part 1, since $\Init(\mathfrak{s}) = (s_1, \dots, s_{k-1})$, we get $s_k > q$. Set $ \begin{pmatrix}
\Sigma  \\
V  \end{pmatrix}  =  \begin{pmatrix}
\varepsilon^{-1} \varepsilon_{k} & \varepsilon_{k+1} &\dots & \varepsilon_n \\
s_k - q & s_{k+1} &\dots & s_n \end{pmatrix} $. By Proposition \ref{polycer1}, $\mathcal{C}_{\Sigma,V}(R_{\varepsilon})$ is of the form
\begin{equation} \label{polyr1}
  \Si_d \begin{pmatrix}
 \varepsilon_{k} & \dots & \varepsilon_n \\
s_{k} & \dots & s_n \end{pmatrix}  +  \Si_d \begin{pmatrix}
 \varepsilon & \varepsilon^{-1}\varepsilon_{k} & \varepsilon_{k+1} & \dots & \varepsilon_n \\
q &  s_k- q & s_{k+1} & \dots & s_n \end{pmatrix}  + \sum \limits_i b_i \Si_{d+1} \begin{pmatrix}
\varepsilon & \bm{\epsilon}_i  \\
1 & \mathfrak{t}_i  \end{pmatrix}  =0,
\end{equation}
where $b_i \in A$ divisible by $D_1$ and $ \begin{pmatrix}
 \bm{\epsilon}_i  \\
\mathfrak{t}_i  \end{pmatrix} $ are positive arrays satisfying for all $i$,
\begin{equation*}
     \begin{pmatrix}
 \bm{\epsilon}_i  \\
\mathfrak{t}_i  \end{pmatrix}  \leq  \begin{pmatrix}
 1  \\
q - 1 \end{pmatrix}  +  \begin{pmatrix}
 \Sigma  \\
V  \end{pmatrix}  =  \begin{pmatrix}
\varepsilon^{-1} \varepsilon_{k} & \varepsilon_{k+1} &\dots & \varepsilon_n \\
s_k - 1 & s_{k+1} &\dots & s_n \end{pmatrix} .
\end{equation*}

For $m \in \mathbb{N}$, we denote by $q^{\{m\}}$ the sequence of length $m$ with all terms equal to $q$. We agree by convention that $q^{\{0\}}$ is the empty sequence. Setting $s_0 = 0$, we may assume that there exists a maximal index $j$ with $0 \leq j \leq k-1$ such that $s_j < q$, hence $\Init(\mathfrak{s}) = (s_1, \dots, s_j, q^{\{k-j-1\}})$.

Then the operator $ \mathcal{BC}_{\varepsilon_{j+1},q} \circ \dots \circ \mathcal{BC}_{\varepsilon_{k-1},q}$ applied to the relation \eqref{polyr1} gives
\begin{align*}
 \Si_d &\begin{pmatrix}
 \varepsilon_{j+1} & \dots & \varepsilon_{k-1} & \varepsilon_{k} & \dots & \varepsilon_n \\
q & \dots & q & s_{k} & \dots & s_n \end{pmatrix}  \\
& +  \Si_d \begin{pmatrix}
\varepsilon_{j+1} & \dots & \varepsilon_{k-1} & \varepsilon & \varepsilon^{-1}\varepsilon_{k} & \epsilon_{k+1} & \dots & \epsilon_n \\
q & \dots & q & q &  s_k- q & s_{k+1} & \dots & s_n \end{pmatrix}  \\
&
+ \sum \limits_i b_{i_1 \dots i_{k-j}} \Si_{d+1} \begin{pmatrix}
\varepsilon_{j+1} & \bm{\epsilon}_{i_1 \dots i_{k-j}}  \\
1 & \mathfrak{t}_{i_1 \dots i_{k-j}}  \end{pmatrix}  =0,
\end{align*}
where $b_{i_1 \dots i_{k-j}} \in A$ are divisible by $D_1$ and for $2 \leq l \leq k-j$, $ \begin{pmatrix}
 \bm{\epsilon}_{i_1 \dots i_l}  \\
 \mathfrak{t}_{i_1 \dots i_l}  \end{pmatrix} $ are positive arrays satisfying 
\begin{equation*}
     \begin{pmatrix}
 \bm{\epsilon}_{i_1 \dots i_l}  \\
 \mathfrak{t}_{i_1 \dots i_l}  \end{pmatrix}  \leq  \begin{pmatrix}
 1  \\
q - 1 \end{pmatrix}  +  \begin{pmatrix}
\varepsilon_{k-l+2} & \bm{\epsilon}_{i_1 \dots i_{l-1}}  \\
1 & \mathfrak{t}_{i_1 \dots i_{l-1}}  \end{pmatrix}  =  \begin{pmatrix}
\varepsilon_{k-l+2} & \bm{\epsilon}_{i_1 \dots i_{l-1}}  \\
q & \mathfrak{t}_{i_1 \dots i_{l-1}}  \end{pmatrix} .
\end{equation*}

Thus 
\begin{equation*} 
     \begin{pmatrix}
 \bm{\epsilon}_{i_1 \dots i_{k-j}}  \\
 \mathfrak{t}_{i_1 \dots i_{k-j}}  \end{pmatrix}  \leq  \begin{pmatrix}
 \varepsilon_{j+2}& \dots & \varepsilon_{k-1} & \varepsilon & \varepsilon^{-1}\varepsilon_{k} & \varepsilon_{k+1}& \dots & \varepsilon_n \\
q& \dots & q &  q & s_{k} - 1 & s_{k+1} &\dots & s_n \end{pmatrix} .
\end{equation*}



Letting $ \begin{pmatrix}
\Sigma'  \\
V'  \end{pmatrix}  =  \begin{pmatrix}
 \varepsilon_{1} &\dots & \varepsilon_j \\
 s_1 &\dots & s_j \end{pmatrix} $, by Proposition \ref{polybesao}, we continue to apply $\mathcal{B}^*_{\Sigma',V'}$ to the above relation and get 
\begin{align*}
  \Si_d \begin{pmatrix}
\bm{\epsilon} \\ \mathfrak{s} \end{pmatrix}  &+  \Si_d \begin{pmatrix}
\varepsilon_1 & \dots & \varepsilon_{k-1} & \varepsilon & \varepsilon^{-1}\varepsilon_{k} & \varepsilon_{k+1} & \dots & \varepsilon_n \\
s_1 & \dots & s_{k-1} & q &  s_k- q & s_{k+1} & \dots & s_n \end{pmatrix}  \\
& + \sum \limits_i b_{i_1 \dots i_{k-j}} \Si_d \begin{pmatrix}
\varepsilon_1 & \dots & \varepsilon_j & \varepsilon_{j+1} & \bm{\epsilon}_{i_1 \dots i_{k-j}}  \\
s_1 & \dots & s_j & 1 & \mathfrak{t}_{i_1 \dots i_{k-j}}  \end{pmatrix}  \\
& + \sum \limits_i b_{i_1 \dots i_{k-j}} \Si_{d+1} \begin{pmatrix}
\varepsilon_1 & \dots & \varepsilon_{j-1} & \varepsilon_j \varepsilon_{j+1} & \bm{\epsilon}_{i_1 \dots i_{k-j}}  \\
s_1 & \dots & s_{j-1} & s_j+1 & \mathfrak{t}_{i_1 \dots i_{k-j}}  \end{pmatrix}  =0.
\end{align*}
 
Hence 
\begin{align*}
  \Li \begin{pmatrix}
\bm{\epsilon} \\ \mathfrak{s} \end{pmatrix}  &+  \Li \begin{pmatrix}
\varepsilon_1 & \dots & \varepsilon_{k-1} & \varepsilon & \varepsilon^{-1}\varepsilon_{k} & \varepsilon_{k+1} & \dots & \varepsilon_n \\
s_1 & \dots & s_{k-1} & q &  s_k- q & s_{k+1} & \dots & s_n \end{pmatrix}  \\
& + \sum \limits_i b_{i_1 \dots i_{k-j}} \Li \begin{pmatrix}
\varepsilon_1 & \dots & \varepsilon_j & \varepsilon_{j+1} & \bm{\epsilon}_{i_1 \dots i_{k-j}}  \\
s_1 & \dots & s_j & 1 & \mathfrak{t}_{i_1 \dots i_{k-j}}  \end{pmatrix}  \\
& + \sum \limits_i b_{i_1 \dots i_{k-j}} \Li \begin{pmatrix}
\varepsilon_1 & \dots & \varepsilon_{j-1} & \varepsilon_j \varepsilon_{j+1} & \bm{\epsilon}_{i_1 \dots i_{k-j}}  \\
s_1 & \dots & s_{j-1} & s_j+1 & \mathfrak{t}_{i_1 \dots i_{k-j}}  \end{pmatrix}  =0.
\end{align*}

The verification of positive arrays of type 1, type 2, type 3 with respect to $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} $ follows the same line as in \cite{ND21}. We omit the details.

For Part 2, the proof follows the same line as in Part 1. We omit this proof.
\end{proof}

We recall the following definition of \cite{ND21} (see \cite[Definition 3.1]{ND21}):
\begin{definition}
Let $k \in \mathbb N$ and $\mathfrak{s}$ be a tuple of positive integers. We say that $\mathfrak{s}$ is $k$-admissible if it satisfies the following two conditions:
\begin{itemize}
\item[1)] $s_1,\dots,s_k \leq q$.

\item[2)] $\mathfrak{s}$ is not of the form $(s_1,\dots,s_r)$ with $r \leq k$, $s_1,\dots,s_{r-1} \leq q$, and $s_r=q$.
\end{itemize}
Here we recall $s_i=0$ for $i > \depth(\mathfrak{s})$.

A positive array is $k$-admissible if the corresponding tuple is $k$-admissible.
\end{definition}

\begin{proposition} \label{polyalgpartlem}
For all $k \in \mathbb{N}$ and for all arrays $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} $, $\Li \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} $ can be expressed as a $K$-linear combination of $\Li \begin{pmatrix}
 \bm{\epsilon}  \\
\mathfrak{t}  \end{pmatrix} $'s of the same weight such that $\mathfrak{t}$ is $k$-admissible.
\end{proposition}

\begin{proof}
The proof follow the same line as that of \cite[Theorem A]{ND21}.
\end{proof}


\fantome{

\begin{proof}
We first denote by $P_k$ the following statement:
"$\Li \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} $ can be expressed as a $K$-linear combination of $\Li \begin{pmatrix}
 \bm{\epsilon}  \\
\mathfrak{t}  \end{pmatrix} $'s of the same weight such that $\mathfrak{t}$ is $k$-admissible."
We need to show that $P_k$ holds for all arrays $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} $ and for all $k \in \mathbb{N}$. We proceed the proof by induction on $k$. 
For $k = 1$, we prove that $P_1$ holds for all arrays $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s} \end{pmatrix} $ by induction on $s_1$. If $s_1 \leq q$, then $\mathfrak{s}$ is $1$-admissible or $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s} \end{pmatrix}  = \begin{pmatrix}
 \varepsilon \\
q \end{pmatrix} $. The proof in the case $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s} \end{pmatrix}  =  \begin{pmatrix}
 \varepsilon \\
q \end{pmatrix} $ follows from the relation $R_{\varepsilon}$. If $s_1 = s >q$, we suppose that $P_1$ holds for all arrays $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s} \end{pmatrix} $ such that $ s_1< s$. We need to show that $P_1$ holds for all arrays $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s} \end{pmatrix} $ such that $s_1=s$. Indeed, we set $ \begin{pmatrix}
\Sigma  \\
V  \end{pmatrix}  =  \begin{pmatrix}
 \varepsilon_1 & \varepsilon_{k+1} &\dots & \varepsilon_n \\
s_1 - q & s_{k+1} &\dots & s_n \end{pmatrix} $. From Proposition \ref{polycer1}, we have $\mathcal{C}_{\Sigma,V}(R_{1})$ is of the form
\begin{equation*} 
  \Si_d \begin{pmatrix}
 \varepsilon_{1} & \dots & \varepsilon_n \\
s_{1} & \dots & s_n \end{pmatrix}  +  \Si_d \begin{pmatrix}
 1 &\varepsilon_1 & \varepsilon_{k+1} &\dots & \varepsilon_n \\
q&  s_1 - q & s_{k+1} &\dots & s_n \end{pmatrix}  + \sum \limits_i b_i \Si_{d+1} \begin{pmatrix}
1 & \epsilon_i  \\
1 & \mathfrak{t}_i  \end{pmatrix}  =0,
\end{equation*}
where $b_i \in K$ and $ \begin{pmatrix}
 \epsilon_i  \\
\mathfrak{t}_i  \end{pmatrix} $ are positive arrays satisfying $ \begin{pmatrix}
 \epsilon_i  \\
\mathfrak{t}_i  \end{pmatrix}  \leq  \begin{pmatrix}
 1  \\
q - 1 \end{pmatrix}  +  \begin{pmatrix}
 \Sigma  \\
V  \end{pmatrix} $ for all $i$. Thus we have 
\begin{equation*} 
  \Li \begin{pmatrix}
 \varepsilon_{1} & \dots & \varepsilon_n \\
s_{1} & \dots & s_n \end{pmatrix}  +  \Li \begin{pmatrix}
 1 &\varepsilon_1 & \varepsilon_{k+1} &\dots & \varepsilon_n \\
q&  s_1 - q & s_{k+1} &\dots & s_n \end{pmatrix}  + \sum \limits_i b_i \Li \begin{pmatrix}
1 & \epsilon_i  \\
1 & \mathfrak{t}_i  \end{pmatrix}  =0,
\end{equation*}
It is obvious that $(q, s_1 - q, s_{k+1}, \dots, s_n)$ and $(1,\mathfrak{t}_i)$ are $1$-admissble, hence $P_1$ holds for $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} $, as desired.
We next suppose that $P_{k-1}$ holds for all arrays $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} $. We prove that $P_k$ holds for all arrays $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} $ by induction on the sum $s_1+\dots + s_k$. The proof in the case $s_1+\dots + s_k = 1$ is a simple check. Assume that $P_k$ holds for all arrays $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} $ such that $s_1 + \dots + s_k < s$. We need to show that $P_k$ holds for all arrays $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} $ such that $s_1 + \dots + s_k = s$. We may assume that $\mathfrak{s}$ is not $k$-admissible. Moreover,  it suffices to consider tuples $\mathfrak{s}$ such that $k \leq \depth(s)$. Otherwise $k-1 \geq \depth(s)$ and the proof follows from the induction hypothesis $P_{k-1}$ holds for all arrays $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} $. We now give an algorithm to show that $P_k$ holds for $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} $. 
\textit{Algorithm:} We begin with an array $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} $ such that $\mathfrak{s}$ is not $k$-admissible, $k \leq \depth(\mathfrak{s})$ and $s_1 + \dots + s_k = s$.
\textit{Step $1$:} As in Proposition \ref{polydecom}, we decompose $\Li \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} $ in the following types:
      \begin{equation}\label{polyt1}
    \Li \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} 
    = \underbrace{ - \Li \begin{pmatrix}
 \bm{\varepsilon}'  \\
\mathfrak{s}'  \end{pmatrix} }_\text{type 1} + \underbrace{\sum\limits_i b_i\Li \begin{pmatrix}
 \epsilon'_i  \\
\mathfrak{t}'_i  \end{pmatrix} }_\text{type 2} + \underbrace{\sum\limits_i c_i\Li \begin{pmatrix}
 \bm{\mu}_i  \\
\mathfrak{u}_i  \end{pmatrix} }_\text{type 3}  ,
    \end{equation}
where $b_i,c_i \in K$.  Note that the term $ - \Li \begin{pmatrix}
 \bm{\varepsilon}' \\
\mathfrak{s}'  \end{pmatrix} $ disappears if $k = \depth(s); s_1, \dots, s_{k-1} \leq q$ and $s_k = q$. It follows from Proposition \ref{polydecom} that for all arrays $ \begin{pmatrix}
 \bm{\epsilon}  \\
\mathfrak{t}  \end{pmatrix} $ appearing in the right hand side of \eqref{polyt1}, we have $k \leq \depth(\mathfrak{s}) \leq \depth(\mathfrak{t})$ and 
\begin{equation*}
    t_1 + \dots + t_k \leq s_1 + \dots + s_k = s.
\end{equation*}
 \textit{Step $2$:} If all arrays $ \begin{pmatrix}
 \bm{\epsilon}  \\
\mathfrak{t}  \end{pmatrix} $ appearing in the right hand side of \eqref{polyt1} are either $k$-admissible or satisfied $t_1 + \dots + t_k <s$, then $P_k$ holds for all $ \begin{pmatrix}
 \bm{\epsilon}  \\
\mathfrak{t}  \end{pmatrix} $ by the induction hypothesis and hence the algorithm ends. If this is not the case, there exists an array $ \begin{pmatrix}
 \bm{\varepsilon}_1  \\
\mathfrak{s}_1  \end{pmatrix} $ such that $\mathfrak{s}_1$ is not $k$-admissible, $k \leq \depth(\mathfrak{s}_1)$ and $s_{11} + \dots + s_{1k} = s$. For all such array $ \begin{pmatrix}
 \bm{\varepsilon}_1  \\
\mathfrak{s}_1  \end{pmatrix} $, we repeat the algorithm for $ \begin{pmatrix}
 \bm{\varepsilon}_1  \\
\mathfrak{s}_1  \end{pmatrix} $.

Note that the array   $ \begin{pmatrix}
 \bm{\varepsilon}'  \\
\mathfrak{s}'  \end{pmatrix} $ of type $1$ and arrays $ \begin{pmatrix}
 \epsilon'  \\
\mathfrak{t}'  \end{pmatrix} $  of type $2$ with respect to $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} $ satisfying
\begin{equation*}
  s'_{1} + \dots + s'_{k} < s_1 + \dots + s_k = s \quad \text{and} \quad  t'_{1} + \dots + t'_{k} < s_1 + \dots + s_k = s
\end{equation*} (see Proposition \ref{polydecom}), hence  arrays $ \begin{pmatrix}
 \bm{\varepsilon}_1  \\
\mathfrak{s}_1  \end{pmatrix} $ appearing in the algorithm is of type $3$ with respect to $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} $.
    It remains to show that the above algorithm ends after finite steps. Suppose in contrast, we thus get a sequence $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  = \begin{pmatrix}
 \bm{\varepsilon}_0  \\
\mathfrak{s}_0  \end{pmatrix} ,  \begin{pmatrix}
 \bm{\varepsilon}_1  \\
\mathfrak{s}_1  \end{pmatrix} ,  \begin{pmatrix}
 \bm{\varepsilon}_2  \\
\mathfrak{s}_2  \end{pmatrix} , \dots$ such that $ \begin{pmatrix}
 \bm{\varepsilon}_{i+1}  \\
\mathfrak{s}_{i+1}  \end{pmatrix} $ is of type $3$ with respect to $ \begin{pmatrix}
 \bm{\varepsilon}_i  \\
\mathfrak{s}_i  \end{pmatrix} $ for all $i \geq 0$. Then we obtain from Proposition \ref{polydecom} a strictly increasing sequence
    \begin{equation}\label{polyt2}
        \Init(\mathfrak{s}) \prec \Init (\mathfrak{s}_1) \prec \Init (\mathfrak{s}_2) \prec \dots.
    \end{equation}
    Since $s_i$ is not $k$-admissible, we get $\depth(\Init(\mathfrak{s}_i)) \leq k$, hence $\Init(\mathfrak{s}_i) \preceq q^{\{k\}}$, contradicting to the sequence \eqref{polyt2}. This shows that the algorithm ends after finite steps, thus proving the proposition. 
\end{proof}

}

\subsubsection{A set of generators $\mathcal{AT}_w$ for ACMPL's}

We recall that $\mathcal{AL}_w$ is the $K$-vector space generated by ACMPL's of weight $w$. We denote by $\mathcal{AT}_w$ the set of all ACMPL's $\Li \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  =  \Li \begin{pmatrix}
 \varepsilon_1 & \dots & \varepsilon_n \\
s_1 & \dots & s_n \end{pmatrix} $ of weight $w$ such that $s_1, \dots, s_{n-1} \leq q$ and $s_n < q$. 

We put $t(w)=|\mathcal{AT}_w|$. Then one verifies that 
\begin{equation*}
    t(w) = \begin{cases}
(q - 1) q^{w-1}& \text{if } 1 \leq w < q, \\
            (q - 1) (q^{w-1} - 1) &\text{if } w = q,
		 \end{cases}
\end{equation*}
and for $w>q$, 
	\[ t(w)=(q-1)\sum \limits_{i = 1}^q t(w-i). \]

We are ready to state a weak version of Brown's theorem for ACMPL's.

\begin{proposition} \label{prop: weak Brown}
The set of all elements $\Li \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} $ such that $\Li \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  \in \mathcal{AT}_w$ forms a set of generators for $\mathcal{AL}_w$.
\end{proposition}
\begin{proof}
The result follows immediately from Proposition \ref{polyalgpartlem} in the case of $k = w$.
\end{proof}

\subsection{A strong version of Brown's theorem for ACMPL's} \label{sec:strong Brown}

\subsubsection{Another set of generators $\mathcal{AS}_w$ for ACMPL's} \label{sec:AS_w}

We consider the set $\mathcal{J}_w$ consisting of positive tuples $\mathfrak{s} = (s_1, \dots, s_n)$ of weight $w$ such that $s_1, \dots, s_{n-1} \leq q$ and $s_n <q$, together with the set  $\mathcal{J}'_w$ consisting of positive tuples $\mathfrak{s} = (s_1, \dots, s_n)$ of weight $w$ such that $ q \nmid s_i$ for all $i$. Then there is a bijection 
\begin{equation*}
    \iota \colon \mathcal{J}'_w \longrightarrow \mathcal{J}_w
\end{equation*}
given as follows: for each tuple $\mathfrak{s} = (s_1, \dots, s_n) \in \mathcal{J}'_w$, since $q \nmid s_i$, we can write $s_i = h_i q + r_i $ where $0 < r_i < q$ and $h_i \in \mathbb{Z}^{\ge0}$. The image $\iota(\mathfrak s)$ is the tuple 
\begin{equation*}
    \iota(\mathfrak s) = (\underbrace{q, \dots, q}_{\text{$h_1$ times}}, r_1 , \dots, \underbrace{q, \dots, q}_{\text{$h_n$ times}}, r_n).
\end{equation*}
Let $\mathcal{AS}_w$ denote the set of ACMPL's $\Li \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} $ such that $\mathfrak s \in \mathcal{J}'_w$. We note that in general, $\mathcal{AS}_w$ is much smaller than $\mathcal{AT}_w$. The only exceptions are when $q=2$ or $w<q$.

\subsubsection{Cardinality of $\mathcal{AS}_w$.}

We now compute $s(w)=|\mathcal{AS}_w|$. To do so we denote by $\mathcal{AJ}_w$ the set consisting of positive arrays $ \begin{pmatrix}
 \varepsilon_1 & \dots & \varepsilon_n \\
s_1 & \dots & s_n \end{pmatrix} $ of weight $w$ such that $q \nmid s_i$ for all $i$ and by $\mathcal{AJ}^1_w$ the set consisting of positive arrays $ \begin{pmatrix}
 \varepsilon_1 & \dots & \varepsilon_n \\
s_1 & \dots & s_n \end{pmatrix} $ of weight~$w$ such that $s_1, \dots, s_{n-1} \leq q, s_n < q$ and $\varepsilon_i = 1$ whenever $s_i = q$ for $1 \leq i \leq n$. We construct a map 
\begin{equation*}
    \varphi \colon \mathcal{AJ}_w \longrightarrow \mathcal{AJ}^1_w
\end{equation*}
as follows: for each array $ \begin{pmatrix}
 \bm{\varepsilon}\\
\mathfrak{s} \end{pmatrix}  =  \begin{pmatrix}
 \varepsilon_1 & \dots & \varepsilon_n \\
s_1 & \dots & s_n \end{pmatrix}  \in \mathcal{AJ}_w$, since $q \nmid s_i$, we can write $s_i = h_i q + r_i $ where $0 < r_i < q$ and $h_i \in \mathbb{Z}^{\ge0}$. The image $\varphi  \begin{pmatrix}
 \bm{\varepsilon}\\
\mathfrak{s} \end{pmatrix} $ is the array 
\begin{equation*}
    \varphi  \begin{pmatrix}
 \bm{\varepsilon}\\
\mathfrak{s} \end{pmatrix}  = \bigg( \underbrace{\begin{pmatrix}
     1 & \dots & 1\\
     q & \dots & q
     \end{pmatrix}}_\text{$h_1$ times}
     \begin{pmatrix}
     \varepsilon_1 \\
     r_1
     \end{pmatrix} \dots
   \underbrace{\begin{pmatrix}
     1 & \dots & 1\\
     q & \dots & q
     \end{pmatrix}}_\text{$h_n$ times}
      \begin{pmatrix}
     \varepsilon_n  \\
     r_n  
     \end{pmatrix} \bigg) .
\end{equation*}
It is easily seen that $\varphi$ is a bijection, hence $|\mathcal{AS}_w| =|\mathcal{AJ}_w| = |\mathcal{AJ}^1_w|$.
Thus $s(w) = |\mathcal{AJ}^1_w|$. One verifies that 
\begin{equation*}
    s(w) = \begin{cases}
(q - 1) q^{w-1}& \text{if } 1 \leq w < q, \\
            (q - 1) (q^{w-1} - 1) &\text{if } w = q,
		 \end{cases}
\end{equation*}
and for $w>q$, 
	\[ s(w)=(q-1)\sum \limits_{i = 1}^{q-1} s(w-i) + s(w - q). \]


\subsubsection{} We state a strong version of Brown's theorem for ACMPL's.

\begin{theorem} \label{thm: strong Brown}
The set $\mathcal{AS}_w$ forms a set of generators for $\mathcal{AL}_w$. In particular, 
	\[ \dim_K \mathcal{AL}_w \leq s(w). \]
\end{theorem}

\begin{proof}
For each $\Li \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  \in \mathcal{AT}_w$, since $\mathfrak s \in \mathcal{J}_w$ and $\iota \colon \mathcal{J}'_w \rightarrow \mathcal{J}_w$ is a bijection, there exists a tuple $\mathfrak{s}' \in \mathcal{J}'_w$ such that $\mathfrak{s} = \iota(\mathfrak{s}')$. By Proposition \ref{polydecom}  and Proposition \ref{prop: weak Brown}, it follows that there exists a tuple $\bm{\varepsilon}' \in (\mathbb{F}^{\times}_q)^{\depth(\mathfrak{s}')} $ such that  $\Li \begin{pmatrix}
 \bm{\varepsilon}'  \\
\mathfrak{s}'  \end{pmatrix} $ can be expressed as follows
\begin{equation*}
    \Li \begin{pmatrix}
 \bm{\varepsilon}'  \\
\mathfrak{s}'  \end{pmatrix}  = \sum  a_{\bm{\epsilon},\mathfrak t}^{\bm{\varepsilon}',\mathfrak{s}'} \Li \begin{pmatrix}
 \bm{\epsilon}  \\
\mathfrak{t}  \end{pmatrix} ,
\end{equation*}
where $ \begin{pmatrix}
 \bm{\epsilon}  \\
\mathfrak{t}  \end{pmatrix} $ ranges over all elements of $\mathcal{AJ}_w$ and $a_{\bm{\epsilon},\mathfrak t}^{\bm{\varepsilon}',\mathfrak{s}'} \in A$ satisfying
\begin{equation*}
 a_{\bm{\epsilon},\mathfrak t}^{\bm{\varepsilon}',\mathfrak{s}'} \equiv \begin{cases}
			\pm 1  \ (\text{mod } D_1) & \text{if }  \begin{pmatrix}
 \bm{\epsilon}  \\
\mathfrak{t}  \end{pmatrix}  =  \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix},\\
            0 \ \ (\text{mod } D_1) & \text{otherwise}.
		 \end{cases}
\end{equation*}
Note that $\Li \begin{pmatrix}
 \bm{\varepsilon}'  \\
\mathfrak{s}'  \end{pmatrix}  \in \mathcal{AS}_w$. Thus the transition matrix from the set consisting of such $\Li \begin{pmatrix}
 \bm{\varepsilon}'  \\
\mathfrak{s}'  \end{pmatrix} $ as above (we allow repeated elements) to the set consisting of $\Li \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} $ with $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  \in \mathcal{AJ}_w$ is invertible. It then follows again from Proposition \ref{prop: weak Brown} that $\mathcal{AS}_w$ is a set of generators for $\mathcal{AL}_w$, as desired. \end{proof}




\section{Dual $t$-motives and linear independence} \label{sec: dual motives}

We continue with the notation given in the Introduction. Further, letting $t$ be another independent variable, we denote by $\mathbb T$ the Tate algebra in the variable $t$ with coefficients in $\mathbb{C}_\infty$ equipped with the Gauss norm $\lVert . \rVert_\infty$, and by $\mathbb L$ the fraction field of $\mathbb T$.

\subsection{Dual $t$-motives} ${}$\par

We recall the notion of dual $t$-motives due to Anderson  (see \cite[\S 4]{BP20} and \cite[\S 5]{HJ20} for more details). We refer the reader to \cite{And86} for the related notion of $t$-motives.

For $i \in \mathbb Z$ we consider the $i$-fold twisting of $\mathbb{C}_\infty((t))$ defined by
\begin{align*}
\mathbb{C}_\infty((t)) & \rightarrow \mathbb{C}_\infty((t)) \\
f=\sum_j a_j t^j & \mapsto f^{(i)}:=\sum_j a_j^{q^i} t^j.
\end{align*}
We extend $i$-fold twisting to matrices with entries in $\mathbb{C}_\infty((t))$ by twisting entry-wise.

Let $\overline K[t,\sigma]$ be the non-commutative $\overline K[t]$-algebra generated by
the new variable $\sigma$ subject to the relation $\sigma f=f^{(-1)}\sigma$ for all $f \in \overline K[t]$.

\begin{definition}
An effective dual $t$-motive is a $\overline K[t,\sigma]$-module $\mathcal M'$ which is free and finitely generated over $\overline K[t]$ such that for $\ell\gg 0$ we have
	\[(t-\theta)^\ell(\mathcal M'/\sigma \mathcal M') = \{0\}.\]
\end{definition}

We mention that effective dual $t$-motives are called Frobenius modules in \cite{CPY19,CH21,Har21,KL16}. Note that Hartl and Juschka \cite[\S 4]{HJ20} introduced a more general notion of dual $t$-motives. In particular, effective dual $t$-motives are always dual $t$-motives.

Throughout this paper we will always work with effective dual $t$-motives. Therefore, we will sometimes drop the word ``effective" where there is no confusion.

Let $\mathcal M$ and $\mathcal M'$ be two effective dual $t$-motives. Then a morphism of effective dual $t$-motives $\mathcal M \to \mathcal M'$ is just a homomorphism of left $\overline K[t,\sigma]$-modules. We denote by $\mathcal F$ the category of effective dual $t$-motives equipped with the trivial object $\mathbf{1}$.

We say that an object $\mathcal M$ of $\mathcal F$ is given by a matrix $\Phi \in \Mat_r(\overline K[t])$ if $\mathcal M$ is a $\overline K[t]$-module free of rank $r$ and the action of $\sigma$ is represented by the matrix $\Phi$ on a given  $\overline K[t]$-basis for $\mathcal M$. We say that an object $\mathcal M$ of $\mathcal F$ is uniformizable or rigid analytically trivial if there exists a matrix $\Psi \in \text{GL}_r(\mathbb T)$ satisfying $\Psi^{(-1)}=\Phi \Psi$. The matrix $\Psi$ is called a rigid analytic trivialization of $\mathcal M$. 

We now recall the Anderson-Brownawell-Papanikolas criterion which is crucial in the sequel (see \cite[Theorem 3.1.1]{ABP04}).

\begin{theorem}[Anderson-Brownawell-Papanikolas] \label{thm:ABP}
Let $\Phi \in \Mat_\ell(\overline K[t])$ be a matrix such that $\det \Phi=c(t-\theta)^s$ for some $c \in \overline K$ and $s \in \mathbb Z^{\geq 0}$. Let $\psi \in \Mat_{\ell \times 1}(\mathcal E)$ be a vector  satisfying $\psi^{(-1)}=\Phi \psi$ and $\rho \in \Mat_{1 \times \ell}(\overline K)$ such that $\rho \psi(\theta)=0$. Then there exists a vector $P \in \Mat_{1 \times \ell}(\overline K[t])$ such that
	\[ P \psi=0 \quad \text{and} \quad P(\theta)=\rho. \]
\end{theorem} 

\subsection{Some constructions of dual $t$-motives} \label{subsec:dual motives} ${}$\par

\subsubsection{General case}

We briefly review some constructions of dual $t$-motives introduced in \cite{CPY19} (see also \cite{Cha14,CH21,Har21}).  Let $\mathfrak{s}=(s_1,\ldots,s_r) \in \mathbb N^r$ be a tuple of positive integers and $\mathfrak{Q}=(Q_1,\dots,Q_r) \in \overline K[t]^r$ satisfying the condition
\begin{equation} \label{eq: condition for Q}
(\lVert Q_1 \rVert_\infty / |\theta|_\infty^{\frac{qs_1}{q-1}})^{q^{i_1}} \ldots (\lVert Q_r \rVert_\infty / |\theta|_\infty^{\frac{qs_r}{q-1}})^{q^{i_r}} \to 0
\end{equation}
as $0  \leq i_r < \dots < i_1$ and $ i_1 \to \infty$.

We consider the dual $t$-motives $\mathcal M_{\mathfrak{s},\mathfrak{Q}}$ and $\mathcal M_{\mathfrak{s},\mathfrak{Q}}'$ attached to $(\mathfrak{s},\mathfrak{Q})$ given by the matrices
\begin{align*}
\Phi_{\mathfrak{s},\mathfrak{Q}} &=
\begin{pmatrix}
(t-\theta)^{s_1+\dots+s_r} & 0 & 0 & \dots & 0 \\
Q_1^{(-1)} (t-\theta)^{s_1+\dots+s_r} & (t-\theta)^{s_2+\dots+s_r} & 0 & \dots & 0 \\
0 & Q_2^{(-1)} (t-\theta)^{s_2+\dots+s_r} & \ddots & & \vdots \\
\vdots & & \ddots & (t-\theta)^{s_r} & 0 \\
0 & \dots & 0 & Q_r^{(-1)} (t-\theta)^{s_r} & 1
\end{pmatrix} \\
&\in \Mat_{r+1}(\overline K[t]),
\end{align*}
and $\Phi'_{\mathfrak{s},\mathfrak{Q}} \in \Mat_r(\overline K[t])$ is the upper left $r \times r$ sub-matrix of $\Phi_{\mathfrak{s},\mathfrak{Q}}$. 

Throughout this paper, we work with the Carlitz period $\widetilde \pi$ which is a fundamental period of the Carlitz module (see \cite{Gos96, Tha04}). We fix a choice of $(q-1)$st root of $(-\theta)$ and set
	\[ \Omega(t):=(-\theta)^{-q/(q-1)} \prod_{i \geq 1} \left(1-\frac{t}{\theta^{q^i}} \right) \in \mathbb T^\times \]
so that
	\[ \Omega^{(-1)}=(t-\theta)\Omega \quad \text{ and } \quad \frac{1}{\Omega(\theta)}=\widetilde \pi. \]
Given $(\mathfrak{s},\mathfrak{Q})$ as above, Chang introduced the following series (see \cite[Lemma 5.3.1]{Cha14} and also \cite[Eq. (2.3.2)]{CPY19})
\begin{align} \label{eq: series L}
\mathfrak{L}(\mathfrak{s};\mathfrak{Q})=\mathfrak{L}(s_1,\dots,s_r;Q_1,\dots,Q_r):=\sum_{i_1 > \dots > i_r \geq 0} (\Omega^{s_r} Q_r)^{(i_r)} \dots (\Omega^{s_1} Q_1)^{(i_1)}.
\end{align}

If we denote $\mathcal E$ the ring of series $\sum_{n \geq 0} a_nt^n \in \overline K[[t]]$ such that $\lim_{n \to +\infty} \sqrt[n]{|a_n|_\infty}=0$ and $[K_\infty(a_0,a_1,\ldots):K_\infty]<\infty$, then any $f \in \mathcal E$ is an entire function. It is proved that $\mathfrak{L}(\mathfrak{s},\mathfrak{Q}) \in \mathcal E$ (see \cite[Lemma 5.3.1]{Cha14}).  In the sequel, we will use the following crucial property of this series (see \cite[Lemma 5.3.5]{Cha14} and \cite[Proposition 2.3.3]{CPY19}): for all $j \in \mathbb Z^{\geq 0}$, we have
\begin{equation} \label{eq: power seriesChang}
\mathfrak{L}(\mathfrak{s};\mathfrak{Q}) \left(\theta^{q^j} \right)=\left(\mathfrak{L}(\mathfrak{s};\mathfrak{Q})(\theta)\right)^{q^j}.
\end{equation}

Then the matrix given by	
\begin{align*}
\Psi_{\mathfrak{s},\mathfrak{Q}} &=
\begin{pmatrix}
\Omega^{s_1+\dots+s_r} & 0 & 0 & \dots & 0 \\
\mathfrak{L}(s_1;Q_1) \Omega^{s_2+\dots+s_r} & \Omega^{s_2+\dots+s_r} & 0 & \dots & 0 \\
\vdots & \mathfrak{L}(s_2;Q_2) \Omega^{s_3+\dots+s_r} & \ddots & & \vdots \\
\vdots & & \ddots & \ddots & \vdots \\
\mathfrak{L}(s_1,\dots,s_{r-1};Q_1,\dots,Q_{r-1}) \Omega^{s_r}  & \mathfrak{L}(s_2,\dots,s_{r-1};Q_2,\dots,Q_{r-1}) \Omega^{s_r} & \dots & \Omega^{s_r}& 0 \\
\mathfrak{L}(s_1,\dots,s_r;Q_1,\dots,Q_r)  & \mathfrak{L}(s_2,\dots,s_r;Q_2,\dots,Q_r) & \dots & \mathfrak{L}(s_r;Q_r) & 1
\end{pmatrix} \\
&\in \text{GL}_{r+1}(\mathbb T)
\end{align*}
satisfies
	\[ \Psi_{\mathfrak{s},\mathfrak{Q}}^{(-1)}=\Phi_{\mathfrak{s},\mathfrak{Q}} \Psi_{\mathfrak{s},\mathfrak{Q}}. \]
Thus $\Psi_{\mathfrak{s},\mathfrak{Q}}$ is a rigid analytic trivialization associated to the dual $t$-motive $\mathcal M_{\mathfrak{s},\mathfrak{Q}}$.

We also denote by $\Psi_{\mathfrak{s},\mathfrak{Q}}'$ the upper $r \times r$ sub-matrix of $\Psi_{\mathfrak{s},\mathfrak{Q}}$. It is clear that $\Psi_\mathfrak{s}'$ is a rigid analytic trivialization associated to the dual $t$-motive $\mathcal M_{\mathfrak{s},\mathfrak{Q}}'$.

Further, combined with Eq. \eqref{eq: power seriesChang}, the above construction of dual $t$-motives implies that $\widetilde \pi^w \frak L(\mathfrak{s};\mathfrak{Q})(\theta)$ where $w=s_1+\dots+s_r$ has the MZ (multizeta) property in the sense of \cite[Definition 3.4.1]{Cha14}. By \cite[Proposition 4.3.1]{Cha14}, we get

\begin{proposition} \label{prop: MZ property}
Let $(\mathfrak{s}_i;\mathfrak{Q}_i)$ as before for $1 \leq i \leq m$. We suppose that all the tuples of positive integers $\mathfrak{s}_i$ have the same weight, says $w$. Then the following assertions are equivalent:
\begin{itemize}
\item[i)] $\frak L(\mathfrak{s}_1;\mathfrak{Q}_1)(\theta),\dots,\frak L(\mathfrak{s}_m;\mathfrak{Q}_m)(\theta)$ are $K$-linearly independent.

\item[ii)] $\frak L(\mathfrak{s}_1;\mathfrak{Q}_1)(\theta),\dots,\frak L(\mathfrak{s}_m;\mathfrak{Q}_m)(\theta)$ are $\overline K$-linearly independent.
\end{itemize}
\end{proposition}

We end this section by mentionning that Chang \cite{Cha14} also proved analogue of Goncharov's conjecture in this setting.

\subsubsection{Dual $t$-motives connected to MZV's and AMZV's} \label{sec:MZV motives} ${}$\par

Following Anderson and Thakur \cite{AT09} we introduce dual $t$-motives connected to MZV's and AMZV's. We briefly review Anderson-Thakur polynomials introduced in \cite{AT90}. For $k \geq 0$, we set $[k]:=\theta^{q^k}-\theta$ and $D_k:= \prod^k_{\ell=1} [\ell]^{q^{k-\ell}}$. For $n \in \ensuremath \mathbb{N}$ we write $n-1 = \sum_{j \geq 0} n_j q^j$ with $0 \leq n_j \leq q-1$ and define
	\[ \Gamma_n:=\prod_{j \geq 0} D_j^{n_j}. \]

We set $\gamma_0(t) :=1$ and $\gamma_j(t) :=\prod^j_{\ell=1} (\theta^{q^j}-t^{q^\ell})$ for $j\geq 1$. Then Anderson-Thakur polynomials $\alpha_n(t) \in A[t]$ are given by the generating series
	\[ \sum_{n \geq 1} \frac{\alpha_n(t)}{\Gamma_n} x^n:=x\left( 1-\sum_{j \geq 0} \frac{\gamma_j(t)}{D_j} x^{q^j} \right)^{-1}. \]
Finally, we define $H_n(t)$ by switching $\theta$ and $t$
\begin{align*}
H_n(t)=\alpha_n(t) \big|_{t=\theta, \, \theta=t}.
\end{align*}
By \cite[Eq. (3.7.3)]{AT90} we get
\begin{equation} \label{eq: deg theta Hn}
\deg_\theta H_n \leq \frac{(n-1) q}{q-1}<\frac{nq}{q-1}.
\end{equation}

Let $\mathfrak{s}=(s_1,\ldots,s_r) \in \mathbb N^r$ be a tuple and $\mathfrak \epsilon=(\epsilon_1,\ldots,\epsilon_r) \in (\mathbb{F}_q^\times)^r$.  For all $1 \leq i \leq r$ we fix a fixed $(q-1)$-th root $\gamma_i$ of $\epsilon_i \in \mathbb F_q^\times$ and set $Q_{s_i,\epsilon_i}:=\gamma_i H_{s_i}$. Then we set $\mathfrak{Q}_{\mathfrak{s},\bm{\epsilon}}:=(Q_{s_1,\epsilon_1},\dots,Q_{s_r,\epsilon_r})$ and put $\frak L(\mathfrak{s};\bm{\epsilon}):=\frak L(\mathfrak{s};\mathfrak{Q}_{\mathfrak{s},\bm{\epsilon}})$. By \eqref{eq: deg theta Hn} we know that $\lVert H_n \rVert_\infty < |\theta|_\infty ^{\tfrac{n q}{q-1}}$ for all $n \in \ensuremath \mathbb{N}$, thus $\mathfrak{Q}_{\mathfrak{s},\bm{\epsilon}}$ satisfies Condition \eqref{eq: condition for Q}. Thus we can define the dual $t$-motives $\mathcal M_{\mathfrak{s},\bm{\epsilon}}=\mathcal M_{\mathfrak{s},\mathfrak{Q}_{\mathfrak{s},\bm{\epsilon}}}$ and $\mathcal M_{\mathfrak{s},\bm{\epsilon}}'=\mathcal M_{\mathfrak{s},\mathfrak{Q}_{\mathfrak{s},\bm{\epsilon}}}'$ attached to $\mathfrak{s}$ whose matrices and rigid analytic trivializations will be denoted by $(\Phi_{\mathfrak{s},\bm{\epsilon}},\Psi_{\mathfrak{s},\bm{\epsilon}})$ and $(\Phi_{\mathfrak{s},\bm{\epsilon}}',\Psi_{\mathfrak{s},\bm{\epsilon}}')$, respectively. These dual $t$-motives are connected to MZV's and AMZV's by the following result (see \cite[Proposition 2.12]{CH21} for more details):
\begin{equation} \label{eq: MZV}
\frak L(\mathfrak{s};\bm{\epsilon})(\theta)=\frac{\gamma_1 \dots \gamma_r \Gamma_{s_1} \dots \Gamma_{s_r} \zeta_A \begin{pmatrix}
\bm{\epsilon} \\ \mathfrak{s}
\end{pmatrix}}{\widetilde \pi^{w(\mathfrak{s})}}.
\end{equation}
By a result of Thakur \cite{Tha09}, one can show (see \cite[Theorem 2.1]{Har21}) that $\zeta_A \begin{pmatrix}
\bm{\epsilon} \\ \mathfrak{s}
\end{pmatrix} \neq 0$. Thus $\frak L(\mathfrak{s};\bm{\epsilon})(\theta) \neq 0$.

\subsubsection{Dual $t$-motives connected to CMPL's and ACMPL's} \label{sec:CMPL motives} ${}$\par

We keep the notation as above. Let $\mathfrak{s}=(s_1,\ldots,s_r) \in \mathbb N^r$ be a tuple and $\bm{\epsilon}=(\epsilon_1,\ldots,\epsilon_r) \in (\mathbb{F}_q^\times)^r$.  For all $1 \leq i \leq r$ we have a fixed $(q-1)$-th root $\gamma_i$ of $\epsilon_i \in \mathbb F_q^\times$ and set $Q_{s_i,\epsilon_i}':=\gamma_i$. Then we set $\mathfrak{Q}_{\mathfrak{s},\bm{\epsilon}}':=(Q_{s_1,\epsilon_1}',\dots,Q_{s_r,\epsilon_r}')$ and put 
\begin{align} \label{eq: series Li}
\mathfrak{Li}(\mathfrak{s};\bm{\epsilon})=\frak L(\mathfrak{s};\mathfrak{Q}_{\mathfrak{s},\bm{\epsilon}}')=\sum_{i_1 > \dots > i_r \geq 0} (\gamma_{i_r} \Omega^{s_r})^{(i_r)} \dots (\gamma_{i_1} \Omega^{s_1})^{(i_1)}. 
\end{align}

Thus we can define the dual $t$-motives $\mathcal N_{\mathfrak{s},\bm{\epsilon}}=\mathcal N_{\mathfrak{s},\mathfrak{Q}_{\mathfrak{s},\bm{\epsilon}}'}$ and $\mathcal N_{\mathfrak{s},\bm{\epsilon}}'=\mathcal N_{\mathfrak{s},\mathfrak{Q}_{\mathfrak{s},\bm{\epsilon}}'}'$ attached to $(\mathfrak{s},\bm{\epsilon})$. These dual $t$-motives are connected to CMPL's and ACMPL's by the following result (see \cite[Lemma 5.3.5]{Cha14} and \cite[Prop. 2.3.3]{CPY19}): 
\begin{equation} \label{eq: ACMPL}
\mathfrak{Li}(\mathfrak{s};\bm{\epsilon})(\theta)=\frac{\gamma_1 \dots \gamma_r \Li \begin{pmatrix}
\bm{\epsilon} \\ \mathfrak{s}
\end{pmatrix}}{\widetilde \pi^{w(\mathfrak{s})}}.
\end{equation}

\subsection{A result for linear independence}

\subsubsection{Setup}  ${}$\par

Let $w \in \ensuremath \mathbb{N}$ be a positive integer. Let $\{(\mathfrak{s}_i;\mathfrak{Q}_i)\}_{1 \leq i \leq n}$ be a collection of pairs satisfying Condition \eqref{eq: condition for Q} such that $\mathfrak{s}_i$ always has weight $w$. We write $\mathfrak{s}_i=(s_{i1},\dots,s_{i \ell_i}) \in \mathbb N^{\ell_i}$ and $\mathfrak{Q}_i=(Q_{i1},\dots,Q_{i\ell_i}) \in (\mathbb F_q^\times)^{\ell_i}$ so that $s_{i1}+\dots+s_{i \ell_i}=w$. We introduce the set of tuples
	\[ I(\mathfrak{s}_i;\mathfrak{Q}_i):=\{\emptyset,(s_{i1};Q_{i1}),\dots,(s_{i1},\dots,s_{i (\ell_i-1)};Q_{i1},\dots,Q_{i(\ell_i-1)})\},\]
and set
	\[ I:=\cup_i I(\mathfrak{s}_i;\mathfrak{Q}_i). \]

For all $(\frak t;\mathfrak{Q}) \in I$, we set
\begin{equation} \label{eq:f}
f_{\mathfrak t,\mathfrak{Q}}:= \sum_i a_i(t) \mathfrak L(s_{i(k+1)},\dots,s_{i \ell_i};Q_{i(k+1)},\dots,Q_{i \ell_i}),
\end{equation}
where the sum runs through the set of indices $i$ such that $(\mathfrak t;\mathfrak{Q})=(s_{i1},\dots,s_{i k};Q_{i1},\dots,Q_{i k})$ for some $0 \leq k \leq \ell_i-1$. In particular, $f_{\emptyset}= \sum_i a_i(t) \mathfrak L(\mathfrak{s}_i;\mathfrak{Q}_i)$.
	
\subsubsection{Linear independence}  ${}$\par

We are now ready to state the main result of this section.

\begin{theorem} \label{theorem: linear independence}
We keep the above notation. We suppose further that $\{(\mathfrak{s}_i;\mathfrak{Q}_i)\}_{1 \leq i \leq n}$ satisfies the following conditions:
\begin{itemize}
\item[(LW)] For any weight $w'<w$, the values $\frak L(\frak t;\mathfrak{Q})(\theta)$ with $(\frak t;\mathfrak{Q}) \in I$ and $w(\frak t)=w'$ are all $K$-linearly independent. In particular, $\frak L(\frak t;\mathfrak{Q})(\theta)$ is always nonzero.

\item[(LD)] There exist $a \in A$ and $a_i \in A$ for $1 \leq i \leq n$ which are not all zero such that
\begin{equation*}
a+\sum_{i=1}^n a_i \frak L(\mathfrak{s}_i;\mathfrak{Q}_i)(\theta)=0.
\end{equation*}
\end{itemize}

Then for all $(\mathfrak t;\mathfrak{Q}) \in I$, $f_{\mathfrak t,\mathfrak{Q}}(\theta)$ belongs to $K$ where $f_{\mathfrak t,\mathfrak{Q}}$ is given as in \eqref{eq:f}.
\end{theorem}

\begin{remark}
1) Here we note that  LW stands for Lower Weights and LD for Linear Dependence. 

2) In fact, we improve \cite[Theorem B]{ND21} in two directions. First, we lift the restriction on the Anderson-Thakur polynomials and tuples $\mathfrak{s}_i$. Second and more important, we allow an extra term $a$ which is crucial in the sequel. More precisely, in the case of MZV's, while \cite[Theorem B]{ND21} investigates linear relations amongs MZV's of weight $w$, Theorem \ref{theorem: linear independence} studies  linear relations amongs MZV's of weight $w$ and suitable powers of the Carlitz period $\widetilde \pi^w$.
\end{remark}

\begin{proof}
The proof will be divided into two steps.

\medskip
\noindent {\bf Step 1.} We first construct a dual $t$-motive to which we will apply the Anderson-Brownawell-Papanikolas criterion. In what follows we set $a_i(t):=a_i \rvert_{\theta=t} \in \mathbb F_q[t]$.

For each pair $(\mathfrak{s}_i;\mathfrak{Q}_i)$ we have attached to it a matrix $\Phi_{\mathfrak{s}_i,\mathfrak{Q}_i}$. For $\mathfrak{s}_i=(s_{i1},\dots,s_{i \ell_i}) \in \mathbb N^{\ell_i}$ and $\mathfrak{Q}_i=(Q_{i1},\dots,Q_{i\ell_i}) \in (\mathbb F_q^\times)^{\ell_i}$ we recall
	\[ I(\mathfrak{s}_i;\mathfrak{Q}_i)=\{\emptyset,(s_{i1};Q_{i1}),\dots,(s_{i1},\dots,s_{i (\ell_i-1)};Q_{i1},\dots,Q_{(\ell_i-1)})\}, \]
and $I:=\cup_i I(\mathfrak{s}_i;\mathfrak{Q}_i)$.

We now construct a new matrix $\Phi'$ by merging the same rows of $\Phi_{\mathfrak{s}_1,\mathfrak{Q}_1}',\ldots,\Phi_{\mathfrak{s}_n,\mathfrak{Q}_n}'$ as follows. Then the matrix $\Phi'$ will be a matrix indexed by elements of $I$, says $\Phi'=\left(\Phi'_{(\mathfrak t;\mathfrak{Q}),(\mathfrak t';\mathfrak{Q}')}\right)_{(\mathfrak t;\mathfrak{Q}),(\mathfrak t';\mathfrak{Q}') \in I} \in \Mat_{|I|}(\overline K[t])$. For the row which corresponds to the empty pair $\emptyset$ we put
\begin{align*}
\Phi'_{\emptyset,(\mathfrak t';\mathfrak{Q}')}=
\begin{cases}
(t-\theta)^w & \text{if } (\mathfrak t';\mathfrak{Q}')=\emptyset, \\
0 & \text{otherwise}.
\end{cases}
\end{align*}
For the row indexed by $(\mathfrak t;\mathfrak{Q})=(s_{i1},\dots,s_{i j};Q_{i1},\dots,Q_{ij})$ for some $i$ and $1 \leq j \leq \ell_i-1$ we put
\begin{align*}
\Phi'_{(\mathfrak t;\mathfrak{Q}),(\mathfrak t';\mathfrak{Q}')}=
\begin{cases}
(t-\theta)^{w-w(\mathfrak t')} & \text{if } (\mathfrak t';\mathfrak{Q}')=(\mathfrak t;\mathfrak{Q}), \\
Q_{ij}^{(-1)} (t-\theta)^{w-w(\mathfrak t')} & \text{if } (\mathfrak t';\mathfrak{Q}')=(s_{i1},\dots,s_{i (j-1)};Q_{i1},\dots,Q_{i (j-1)}), \\
0 & \text{otherwise}.
\end{cases}
\end{align*}
Note that $\Phi_{\mathfrak{s}_i,\mathfrak{Q}_i}'=\left(\Phi'_{(\mathfrak t;\mathfrak{Q}),(\mathfrak t';\mathfrak{Q}')}\right)_{(\mathfrak t;\mathfrak{Q}),(\mathfrak t';\mathfrak{Q}') \in I(\mathfrak{s}_i;\mathfrak{Q}_i)}$ for all $i$.

We define $\Phi \in \Mat_{|I|+1}(\overline K[t])$ by
\begin{align*}
\Phi=\begin{pmatrix}
\Phi' & 0  \\
\mathbf{v} & 1
\end{pmatrix} \in \Mat_{|I|+1}(\overline K[t]), \quad \mathbf{v}=(v_{\mathfrak t,\mathfrak{Q}})_{(\mathfrak t;\mathfrak{Q}) \in I} \in \Mat_{1 \times|I|}(\overline K[t]),
\end{align*}
where
\begin{align*}
v_{\mathfrak t,\mathfrak{Q}}=
\begin{cases}
a_i(t) Q_{i \ell_i}^{(-1)} (t-\theta)^{w-w(\mathfrak t)} & \text{if } (\mathfrak t;\mathfrak{Q})=(s_{i1},\dots,s_{i (\ell_i-1)};Q_{i1},\dots,Q_{i (\ell_i-1)}), \\
0 & \text{otherwise}.
\end{cases}
\end{align*}

We now introduce a rigid analytic trivialization matrix $\Psi$ for $\Phi$. We define $\Psi'=\left(\Psi'_{(\mathfrak t;\mathfrak{Q}),(\mathfrak t';\mathfrak{Q}')}\right)_{(\mathfrak t;\mathfrak{Q}),(\mathfrak t';\mathfrak{Q}') \in I} \in  \text{GL}_{|I|}(\mathbb T)$ as follows. For the row which corresponds to the empty pair $\emptyset$ we define
\begin{align*}
\Psi'_{\emptyset,(\mathfrak t';\mathfrak{Q}')}=
\begin{cases}
\Omega^w & \text{if } (\mathfrak t';\mathfrak{Q}')=\emptyset, \\
0 & \text{otherwise}.
\end{cases}
\end{align*}
For the row indexed by $(\mathfrak t;\mathfrak{Q})=(s_{i1},\dots,s_{i j};Q_{i1},\dots,Q_{i j})$ for some $i$ and $1 \leq j \leq \ell_i-1$ we put
\begin{align*}
&\Psi'_{(\mathfrak t;\mathfrak{Q}),(\mathfrak t';\mathfrak{Q}')}= \\
&
\begin{cases}
\mathfrak L(\mathfrak t;\mathfrak{Q}) \Omega^{w-w(\mathfrak t)} & \text{if } (\mathfrak t';\mathfrak{Q}')=\emptyset, \\
\mathfrak L(s_{i(k+1)},\dots,s_{ij};Q_{i(k+1)},\dots,Q_{ij}) \Omega^{w-w(\mathfrak t)} & \text{if } (\mathfrak t';\mathfrak{Q}')=(s_{i1},\dots,s_{i k};Q_{i1},\dots,Q_{i k}) \text{ for some } 1 \leq k \leq j, \\
0 & \text{otherwise}.
\end{cases}
\end{align*}
Note that $\Psi_{\mathfrak{s}_i,\mathfrak{Q}_i}'=\left(\Psi'_{(\mathfrak t;\mathfrak{Q}),(\mathfrak t';\mathfrak{Q}')}\right)_{(\mathfrak t;\mathfrak{Q}),(\mathfrak t';\mathfrak{Q}') \in I(\mathfrak{s}_i;\mathfrak{Q}_i)}$ for all $i$.

We define $\Psi \in \text{GL}_{|I|+1}(\mathbb T)$ by
\begin{align*}
\Psi=\begin{pmatrix}
\Psi' & 0  \\
\mathbf{f} & 1
\end{pmatrix} \in \text{GL}_{|I|+1}(\mathbb T), \quad \mathbf{f}=(f_{\mathfrak t,\mathfrak{Q}})_{\mathfrak t \in I} \in \Mat_{1 \times|I|}(\mathbb T).
\end{align*}
Here we recall (see Eq. \eqref{eq:f})
\begin{equation*}
f_{\mathfrak t,\mathfrak{Q}}= \sum_i a_i(t) \mathfrak L(s_{i(k+1)},\dots,s_{i \ell_i};Q_{i(k+1)},\dots,Q_{i \ell_i})
\end{equation*}
where the sum runs through the set of indices $i$ such that $(\mathfrak t;\mathfrak{Q})=(s_{i1},\dots,s_{i k};Q_{i1},\dots,Q_{i k})$ for some $0 \leq k \leq \ell_i-1$. In particular, $f_{\emptyset}= \sum_i a_i(t) \mathfrak L(\mathfrak{s}_i;\mathfrak{Q}_i)$.

By construction and by \S \ref{subsec:dual motives}, we get $\Psi^{(-1)}=\Phi \Psi$, that means $\Psi$ is a rigid analytic trivialization for $\Phi$.

\medskip
\noindent {\bf Step 2.} Next we apply the Anderson-Brownawell-Papanikolas criterion (see Theorem \ref{thm:ABP}) to prove Theorem \ref{theorem: linear independence}.

In fact, we define 
\begin{align*}
\widetilde \Phi=\begin{pmatrix}
1 & 0  \\
0 & \Phi
\end{pmatrix} \in  \Mat_{|I|+2}(\overline K[t])
\end{align*}
and consider the vector constructed from the first column vector of $\Psi$
\begin{align*}
\widetilde \psi=\begin{pmatrix}
1 \\
\Psi_{(\mathfrak t;\mathfrak{Q}),\emptyset}' \\
f_\emptyset
\end{pmatrix}_{(\mathfrak t;\mathfrak{Q}) \in I}.
\end{align*}
Then we have $\widetilde \psi^{(-1)}=\widetilde \Phi \widetilde \psi$.

We also observe that for all $(\mathfrak t;\mathfrak{Q}) \in I$ we have $\Psi_{(\mathfrak t;\mathfrak{Q}),\emptyset}'=\mathfrak L(\mathfrak t;\mathfrak{Q}) \Omega^{w-w(\mathfrak t)}$. Further,
\begin{align*}
a+f_\emptyset(\theta)=a+\sum_i a_i \mathfrak L(\mathfrak{s}_i;\mathfrak{Q}_i)(\theta)=0.
\end{align*}
By Theorem \ref{thm:ABP} with $\rho=(a,0,\dots,0,1)$ we deduce that there exists $\mathbf{h}=(g_0,g_{\mathfrak t,\mathfrak{Q}},g) \in \Mat_{1 \times (|I|+2)}(\overline K[t])$ such that $\mathbf{h} \psi=0$, and that 
$g_{\mathfrak t,\mathfrak{Q}}(\theta)=0$ for $(\mathfrak t,\mathfrak{Q}) \in I$, $g_0(\theta)=a$ and $g(\theta)=1 \neq 0$. If we put $ \mathbf{g}:=(1/g)\mathbf{h} \in \Mat_{1 \times (|I|+2)}(\overline K(t))$, then all the entries of $ \mathbf{g}$ are regular at $t=\theta$.

Now we have
\begin{align} \label{eq: reduction}
( \mathbf{g}- \mathbf{g}^{(-1)} \widetilde \Phi) \widetilde \psi= \mathbf{g} \widetilde \psi-( \mathbf{g}  \widetilde \psi)^{(-1)}=0.
\end{align}
We write $ \mathbf{g}- \mathbf{g}^{(-1)} \widetilde \Phi=(B_0,B_{\mathfrak t},0)_{\mathfrak t \in I}$. We claim that $B_0=0$ and $B_{\mathfrak t,\mathfrak{Q}}=0$ for all $(\mathfrak t;\mathfrak{Q}) \in I$. In fact, expanding \eqref{eq: reduction} we obtain
\begin{equation} \label{eq: B}
B_0+\sum_{\mathfrak t \in I} B_{\mathfrak t,\mathfrak{Q}} \mathfrak L(\mathfrak t;\mathfrak{Q}) \Omega^{w-w(\mathfrak t)}=0.
\end{equation}

By \eqref{eq: power seriesChang} we see that for $(\mathfrak t;\mathfrak{Q}) \in I$ and $j \in \ensuremath \mathbb{N}$,
\begin{equation*}
\mathfrak{L}(\mathfrak t;\mathfrak{Q})(\theta^{q^j})=(\mathfrak{L}(\mathfrak t;\mathfrak{Q})(\theta))^{q^j}
\end{equation*}
which is nonzero by Condition $(LW)$.

First, as the function $\Omega$ has a simple zero at $t=\theta^{q^k}$ for $k \in \ensuremath \mathbb{N}$, specializing \eqref{eq: B} at $t=\theta^{q^j}$ yields $B_0(\theta^{q^j})=0$ for $j \geq 1$. Since $B_0$ belongs to $\overline K(t)$, it follows that $B_0=0$.

Next, we put $w_0:=\max_{(\mathfrak t;\mathfrak{Q}) \in I} w(\mathfrak t)$ and denote by $I(w_0)$ the set of $(\mathfrak t;\mathfrak{Q}) \in I$ such that $w(\mathfrak t)=w_0$. Then dividing \eqref{eq: B} by $\Omega^{w-w_0}$ yields
\begin{equation} \label{eq: B1}
\sum_{(\mathfrak t;\mathfrak{Q}) \in I} B_{\mathfrak t,\mathfrak{Q}} \mathfrak L(\mathfrak t;\mathfrak{Q}) \Omega^{w_0-w(\mathfrak t)}=\sum_{(\mathfrak t;\mathfrak{Q}) \in I(w_0)} B_{\mathfrak t,\mathfrak{Q}} \mathfrak L(\mathfrak t;\mathfrak{Q})+\sum_{(\mathfrak t;\mathfrak{Q}) \in I \setminus I(w_0)} B_{\mathfrak t,\mathfrak{Q}} \mathfrak L(\mathfrak t;\mathfrak{Q}) \Omega^{w_0-w(\mathfrak t)}=0.
\end{equation}
Since each $B_{\mathfrak t,\mathfrak{Q}}$ belongs to $\overline K(t)$, they are defined at $t=\theta^{q^j}$ for $j \gg 1$. Note that the function $\Omega$ has a simple zero at $t=\theta^{q^k}$ for $k \in \ensuremath \mathbb{N}$. Specializing \eqref{eq: B1} at $t=\theta^{q^j}$ and using \eqref{eq: power seriesChang} yields
	\[ \sum_{(\mathfrak t;\mathfrak{Q}) \in I(w_0)} B_{\mathfrak t,\mathfrak{Q}}(\theta^{q^j}) (\mathfrak L(\mathfrak t;\mathfrak{Q})(\theta))^{q^j}=0 \]
for $j \gg 1$.

We claim that $B_{\mathfrak t,\mathfrak{Q}}(\theta^{q^j})=0$ for $j \gg 1$ and for all $(\mathfrak t;\mathfrak{Q}) \in I(w_0)$. Otherwise, we get a non trivial $\overline K$-linear relation among $\mathfrak L(\mathfrak t;\mathfrak{Q})(\theta)$ with $(\frak t;\mathfrak{Q}) \in I$ of weight $w_0$. By Proposition \ref{prop: MZ property} we deduce a non trivial $K$-linear relation among $\mathfrak L(\mathfrak t;\mathfrak{Q})(\theta)$ with $(\frak t;\mathfrak{Q}) \in I(w_0)$, which contradicts with Condition $(LW)$.
Now we know that $B_{\mathfrak t,\mathfrak{Q}}(\theta^{q^j})=0$ for $j \gg 1$ and for all $(\mathfrak t;\mathfrak{Q}) \in I(w_0)$. Since each $B_{\mathfrak t,\mathfrak{Q}}$ belongs to $\overline K(t)$, it follows that
$B_{\mathfrak t,\mathfrak{Q}}=0$ for all $(\mathfrak t;\mathfrak{Q}) \in I(w_0)$.

Next, we put $w_1:=\max_{(\mathfrak t;\mathfrak{Q}) \in I \setminus I(w_0)} w(\mathfrak t)$ and denote by $I(w_1)$ the set of $(\mathfrak t;\mathfrak{Q}) \in I$ such that $w(\mathfrak t)=w_1$. Dividing \eqref{eq: B} by $\Omega^{w-w_1}$ and specializing at $t=\theta^{q^j}$ yields
	\[ \sum_{(\mathfrak t;\mathfrak{Q}) \in I(w_1)} B_{\mathfrak t,\mathfrak{Q}}(\theta^{q^j}) (\mathfrak L(\mathfrak t;\mathfrak{Q})(\theta))^{q^j}=0 \]
for $j \gg 1$. Since $w_1<w$, by Proposition \ref{prop: MZ property} and Condition $(LW)$ again we deduce that $B_{\mathfrak t,\mathfrak{Q}}(\theta^{q^j})=0$ for $j \gg 1$ and for all $(\mathfrak t;\mathfrak{Q}) \in I(w_1)$. Since each $B_{\mathfrak t,\mathfrak{Q}}$ belongs to $\overline K(t)$, it follows that
$B_{\mathfrak t,\mathfrak{Q}}=0$ for all $(\mathfrak t;\mathfrak{Q}) \in I(w_1)$. Repeating the previous arguments we deduce that $B_{\mathfrak t,\mathfrak{Q}}=0$ for all $(\mathfrak t;\mathfrak{Q}) \in I$ as required.

We have proved that $ \mathbf{g}- \mathbf{g}^{(-1)} \widetilde \Phi=0$. Thus
\begin{align*}
\begin{pmatrix}
1 & 0 & 0 \\
0 & \text{Id} & 0 \\
g_0/g & (g_{\mathfrak t,\mathfrak{Q}}/g)_{(\mathfrak t;\mathfrak{Q}) \in I}  & 1 
\end{pmatrix}^{(-1)} 
\begin{pmatrix}
1 & 0  \\
0 & \Phi
\end{pmatrix}
= \begin{pmatrix}
1 & 0 & 0 \\
0 & \Phi' & 0 \\
0 & 0 & 1
\end{pmatrix}
\begin{pmatrix}
1 & 0 & 0 \\
0 & \text{Id} & 0 \\
g_0/g & (g_{\mathfrak t,\mathfrak{Q}}/g)_{(\mathfrak t;\mathfrak{Q}) \in I}  & 1 
\end{pmatrix}. 
\end{align*}
By \cite[Prop. 2.2.1]{CPY19} we see that the common denominator $b$ of $g_0/g$ and $g_{\mathfrak t,\mathfrak{Q}}/g$ for $(\mathfrak t,\mathfrak{Q}) \in I$ belongs to $\mathbb F_q[t] \setminus \{0\}$. If we put $\delta_0=bg_0/g$ and $\delta_{\mathfrak t,\mathfrak{Q}}=b g_{\mathfrak t,\mathfrak{Q}}/g$ for $(\mathfrak t,\mathfrak{Q}) \in I$ which belong to $\overline K[t]$ and $\delta:=(\delta_{\mathfrak t,\mathfrak{Q}})_{\mathfrak t \in I} \in \Mat_{1 \times |I|}(\overline K[t])$, then $\delta_0^{(-1)}=\delta_0$ and 
\begin{align} \label{eq:equation for delta}
\begin{pmatrix}
\text{Id} & 0 \\
\delta & 1
\end{pmatrix}^{(-1)} 
\begin{pmatrix}
\Phi' & 0  \\
b \mathbf{v} & 1 
\end{pmatrix}
= \begin{pmatrix}
 \Phi' & 0 \\
0 & 1
\end{pmatrix}
\begin{pmatrix}
\text{Id} & 0 \\
\delta & 1
\end{pmatrix}. 
\end{align}

If we put $X:=\begin{pmatrix}
\text{Id} & 0 \\
\delta & 1
\end{pmatrix} 
\begin{pmatrix}
\Psi' & 0  \\
b \mathbf{f} & 1 
\end{pmatrix}$, then $X^{(-1)}=\begin{pmatrix}
 \Phi' & 0 \\
0 & 1
\end{pmatrix} X$. By \cite[\S 4.1.6]{Pap08} there exist $\nu_{\mathfrak t,\mathfrak{Q}} \in \mathbb F_q(t)$ for $(\mathfrak t,\mathfrak{Q}) \in I$ such that if we set $\nu=(\nu_{\mathfrak t,\mathfrak{Q}})_{(\mathfrak t,\mathfrak{Q}) \in I} \in \Mat_{1 \times |I|}(\mathbb F_q(t))$, 
\begin{align*}
X=\begin{pmatrix}
\Psi' & 0 \\
0 & 1
\end{pmatrix} 
\begin{pmatrix}
\text{Id} & 0 \\
\nu & 1
\end{pmatrix}.
\end{align*}
Thus the equation $\begin{pmatrix}
\text{Id} & 0 \\
\delta & 1
\end{pmatrix} 
\begin{pmatrix}
\Psi' & 0  \\
b \mathbf{f} & 1 
\end{pmatrix}=\begin{pmatrix}
\Psi' & 0 \\
0 & 1
\end{pmatrix} 
\begin{pmatrix}
\text{Id} & 0 \\
\nu & 1
\end{pmatrix}$ implies 
\begin{equation} \label{eq:nu}
\delta \Psi'+b \mathbf{f}=\nu. 
\end{equation}
The left-hand side belongs to $\mathbb T$, so does the right-hand side. Thus $\nu=(\nu_{\mathfrak t,\mathfrak{Q}})_{(\mathfrak t,\mathfrak{Q}) \in I} \in \Mat_{1 \times |I|}(\mathbb F_q[t])$. For any $j \in \ensuremath \mathbb{N}$, by specializing \eqref{eq:nu} at $t=\theta^{q^j}$ and using \eqref{eq: power seriesChang} and the fact that $\Omega$ has a simple zero at $t=\theta^{q^j}$ we deduce that
	\[ \mathbf{f}(\theta)=\nu(\theta)/b(\theta). \]
Thus for all $(\mathfrak t,\mathfrak{Q}) \in I$, $f_{\mathfrak t,\mathfrak{Q}}(\theta)$ given as in \eqref{eq:f} belongs to $K$. The proof is complete.
\end{proof}


\section{Linear relations among ACMPL's} \label{sec:transcendental part}

In this section we use freely the notation of \S \ref{sec:algebraic part} and  \S \ref{sec:CMPL motives}.

\subsection{Preliminaries} ${}$\par

We begin this section by proving several auxiliary lemmas which will be useful in the sequel.

\begin{lemma} \label{lem: gamma_i}
Let $\epsilon_i \in \mathbb F_q^\times$ be different elements. We denote by $\gamma_i \in \overline{\mathbb{F}}_q$ a $(q-1)$-th root of $\epsilon_i$. Then $\gamma_i$ are all $\mathbb F_q$-linearly independent.
\end{lemma}

\begin{proof}
We know that $\mathbb F_q^\times$ is cyclic as a multiplicative group. Let $\epsilon$ be a generating element of  $\mathbb F_q^\times$ so that $\mathbb F_q^\times=\langle \epsilon \rangle$. Let $\gamma$ be the associated $(q-1)$-th root of $\epsilon$. Then for all $1 \leq i \leq q-1$ it follows that $\gamma^i$ is a $(q-1)$-th root of $\epsilon^i$. Thus it suffices to show that the polynomial $P(X)=X^{q-1}-\epsilon$ is irreducible in $\mathbb F_q[X]$. Suppose that this is not the case, write $P(X)=P_1(X)P_2(X)$ with $1 \leq \deg P_1<q-1$. Since the roots of $P(X)$ are of the form $\alpha \gamma$ with $\alpha \in \mathbb F_q^\times$, those of $P_1(X)$ are also of this form. Looking at the constant term of $P_1(X)$, we deduce that $\gamma^{\deg P_1} \in \mathbb F_q^\times$. If we put $m=\text{pgcd}(\deg P_1,q-1)$, then $1 \leq m<q-1$ and $\gamma^m \in \mathbb F_q^\times$. Letting $\beta:=\gamma^m \in \mathbb F_q^\times$, we get $\beta^{\frac{q-1}{m}}=\gamma^{q-1}=\epsilon$. Since $1 \leq m<q-1$, we get a contradiction with the fact that $\mathbb F_q^\times=\langle \epsilon \rangle$. The proof is finished.
\end{proof}

\begin{lemma} \label{lem: same character}
Let $ \Li \begin{pmatrix}
 \bm{\epsilon}_i  \\
\mathfrak{s}_i \end{pmatrix} \in \mathcal{AL}_w$ and $a_i \in K$ satisfying
\begin{equation*}
\sum_i a_i \mathfrak{Li}(\mathfrak{s}_i;\bm{\epsilon}_i)(\theta)=0.
\end{equation*}
For $\epsilon \in \mathbb F_q^\times$ we denote by $I(\epsilon)=\{i:\, \chi(\epsilon_i)=\epsilon\}$ the set of pairs such that the corresponding character equals $\epsilon$. Then for all $\epsilon \in \mathbb F_q^\times$,
	\[ \sum_{i \in I(\epsilon)} a_i \mathfrak{Li}(\mathfrak{s}_i;\bm{\epsilon}_i)(\theta)=0. \]
\end{lemma}

\begin{proof}
We keep the notation of Lemma \ref{lem: gamma_i}. Suppose that we have a relation
	\[\sum_i \gamma_i a_i=0\]
with $a_i \in K_\infty$. By Lemma \ref{lem: gamma_i} and the fact that $K_\infty=\mathbb F_q((1/\theta))$, we deduce that $a_i=0$ for all $i$.

By \eqref{eq: MZV} the relation $\sum_i a_i \mathfrak{Li}(\mathfrak{s}_i;\bm{\epsilon}_i)(\theta)=0$ is equivalent to the following one
	\[ \sum_i a_i \gamma_{i1} \dots \gamma_{i\ell_i}  \Li \begin{pmatrix}
 \bm{\epsilon}_i  \\
\mathfrak{s}_i  \end{pmatrix}=0. \]
By the previous discussion, for all $\epsilon \in \mathbb F_q^\times$,
	\[ \sum_{i \in I(\epsilon)} a_i \gamma_{i1} \dots \gamma_{i\ell_i}  \Li \begin{pmatrix}
 \bm{\epsilon}_i  \\
\mathfrak{s}_i  \end{pmatrix}=0. \]
By \eqref{eq: MZV} again we deduce the desired relation	
	\[ \sum_{i \in I(\epsilon)} a_i \mathfrak{Li}(\mathfrak{s}_i;\bm{\epsilon}_i)(\theta)=0. \]
\end{proof}

\begin{lemma} \label{lem: KuanLin}
Let $m \in \mathbb N$, $\varepsilon \in \mathbb F_q^\times$, $\delta \in \overline K[t]$ and $F(t,\theta) \in \overline{\mathbb{F}}_q[t,\theta]$ (resp. $F(t,\theta) \in \mathbb{F}_q[t,\theta]$) satisfying
	\[ \varepsilon\delta = \delta^{(-1)}(t-\theta)^m+F^{(-1)}(t,\theta). \]
Then $\delta \in \overline{\mathbb{F}}_q[t,\theta]$ (resp. $\delta \in \mathbb{F}_q[t,\theta]$) and
	\[ \deg_\theta \delta \leq \max\left\{\frac{qm}{q-1},\frac{\deg_\theta F(t,\theta)}{q}\right\}. \]
\end{lemma}

\begin{proof}
The proof follows the same line as that of \cite[Theorem 2]{KL16} where it is shown that if $F(t,\theta) \in \mathbb{F}_q[t,\theta]$ and $\varepsilon=1$, then $\delta \in \mathbb{F}_q[t,\theta]$. We write down the proof for the case $F(t,\theta) \in \overline{\mathbb{F}}_q[t,\theta]$ for the convenience of the reader.

By twisting once the equality $\varepsilon\delta = \delta^{(-1)}(t-\theta)^m+F^{(-1)}(t,\theta)$ and the fact that $\varepsilon^q=\varepsilon$, we get
	\[ \varepsilon\delta^{(1)} = \delta(t-\theta^q)^m+F(t,\theta). \]
We put $n=\deg_t \delta$ and express
	\[ \delta=a_n t^n + \dots +a_1t+a_0 \in \overline K[t] \]
with $a_0,\dots,a_n \in \overline K$. For $i<0$ we put $a_i=0$.

Since $\deg_t \delta^{(1)}=\deg_t \delta=n < \delta(t-\theta^q)^m=n+m$, it follows that $\deg_t F(t,\theta)=n+m$. Thus we write $F(t,\theta)=b_{n+m} t^{n+m}+\dots+b_1 t+b_0$ with $b_0,\dots,b_{n+m} \in \overline{\mathbb{F}}_q[\theta]$. Plugging into the previous equation, we obtain
	\[ \varepsilon(a_n^q t^n + \dots +a_0^q) = (a_n t^n + \dots +a_0)(t-\theta^q)^m+b_{n+m} t^{n+m}+\dots+b_0. \]

Comparing the coefficients $t^j$ for $n+1 \leq j \leq n+m$ yields
	\[ a_{j-m}+\sum_{i=j-m+1}^{n} {m \choose j-i} (-\theta^q)^{m-j+i} a_i+b_j=0. \]
Since $b_j \in \overline{\mathbb{F}}_q[\theta]$ for all $n+1 \leq j \leq n+m$, we can show by descending induction that $a_j \in \overline{\mathbb{F}}_q[\theta]$ for all $n+1-m \leq j \leq n$.

If $n+1-m \leq 0$, then we are done. Otherwise, comparing the coefficients $t^j$ for $m \leq j \leq n$ yields
	\[ a_{j-m}+\sum_{i=j-m+1}^{n} {m \choose j-i} (-\theta^q)^{m-j+i} a_i+b_j-\varepsilon a_j^q=0. \]
Since $b_j \in \overline{\mathbb{F}}_q[\theta]$ for all $m \leq j \leq n$ and $a_j \in \overline{\mathbb{F}}_q[\theta]$ for all $n+1-m \leq j \leq n$, we can show by descending induction that $a_j \in \overline{\mathbb{F}}_q[\theta]$ for all $0 \leq j \leq n-m$. We conclude that $\delta \in \overline{\mathbb{F}}_q[t,\theta]$.

We now show that $\deg_\theta \delta \leq \max\{\frac{qm}{q-1},\frac{\deg_\theta F(t,\theta)}{q}\}$. Otherwise, suppose that $\deg_\theta \delta > \max\{\frac{qm}{q-1},\frac{\deg_\theta F(t,\theta)}{q}\}$. Then $\deg_\theta \delta^{(1)}=q \deg_\theta \delta$. It implies that $\deg_\theta \delta^{(1)} > \deg_\theta (\delta(t-\theta^q)^m)=\deg_\theta \delta+qm$ and $\deg_\theta \delta^{(1)} >\deg_\theta F(t,\theta)$. Hence we get
	\[\deg_\theta (\varepsilon \delta^{(1)})= \deg_\theta \delta^{(1)} > \deg_\theta(\delta(t-\theta^q)^m+F(t,\theta)), \]
which is a contradiction.
\end{proof}

\subsection{Linear relations: statement of the main result}

\begin{theorem} \label{thm: trans ACMPL}
Let $w \in \ensuremath \mathbb{N}$. We recall that the set  $\mathcal{J}'_w$ consists of positive tuples $\mathfrak{s} = (s_1, \dots, s_n)$ of weight $w$ such that $ q \nmid s_i$ for all $i$. Suppose that we have a non trivial relation
\begin{equation*} 
a+\sum_{\mathfrak{s}_i \in \mathcal{J}'_w} a_i \mathfrak{Li}(\mathfrak{s}_i;\bm{\epsilon}_i)(\theta)=0, \quad \text{for $a, a_i \in K$.}
\end{equation*}
Then $q-1 \mid w$ and $a \neq 0$.

Further, if $q-1 \mid w$, then there is a unique relation
\begin{equation*} 
1+\sum_{\mathfrak{s}_i \in \mathcal{J}'_w} a_i \mathfrak{Li}(\mathfrak{s}_i;\bm{\epsilon}_i)(\theta)=0, \quad \text{for $a_i \in K$.}
\end{equation*}

In particular, the ACMPL's in $\mathcal{AS}_w$ are linearly independent over $K$.
\end{theorem}

\begin{remark} \label{rem:remark on Brown}
We stress that although Theorem \ref{thm: trans ACMPL} is purely a transcendental result, it is crucial that we do need the full strength of the algebraic theory for ACMPL's (i.e., Theorem \ref{thm: strong Brown}) to conclude (see the last step of the proof).  
\end{remark}

As a direct consequence of Theorem \ref{thm: trans ACMPL}, we obtain:
\begin{theorem} \label{thm:ACMPL}
Let $w \in \ensuremath \mathbb{N}$. Then the ACMPL's in $\mathcal{AS}_w$ form a basis for $\mathcal{AL}_w$. In particular,
	\[ \dim_K \mathcal{AL}_w=s(w). \]
\end{theorem}

\begin{proof}
By Theorem \ref{thm: trans ACMPL} the ACMPL's in $\mathcal{AS}_w$ are all linearly independent over $K$. Then by Theorem \ref{thm: strong Brown} we deduce that the ACMPL's in $\mathcal{AS}_w$ form a basis for $\mathcal{AL}_w$. Hence $\dim_K \mathcal{AL}_w=|\mathcal{AL}_w|=s(w)$ as required.
\end{proof}

\subsection{Proof of Theorem \ref{thm: trans ACMPL}}

It is clear that if $q-1 \nmid w$, then any linear relation
	\[ a+\sum_{\mathfrak{s}_i \in \mathcal{J}'_w} a_i \mathfrak{Li}(\mathfrak{s}_i;\bm{\epsilon}_i)(\theta)=0 \]
with $a,a_i \in K$ implies that $a=0$.

The proof is by induction on the weight $w \in \ensuremath \mathbb{N}$. For $w=1$, by the previous remark it suffices to show that if 
	\[a+\sum_i a_i \mathfrak{Li}(1;\epsilon_i)(\theta)=0, \]
then $a_i=0$ for all $i$. In fact, it follows immediately from Lemma \ref{lem: same character}. Suppose that Theorem \ref{thm: trans ACMPL} holds for all $w'<w$. We now prove that it holds for $w$. Suppose that we have a linear relation
\begin{equation} \label{eq: non trivial relation MZV}
a+\sum_i a_i \mathfrak{Li}(\mathfrak{s}_i;\bm{\epsilon}_i)(\theta)=0.
\end{equation}
By Lemma \ref{lem: same character} we can suppose further that $\bm{\epsilon}_i$ has the same character, i.e., there exists $\epsilon \in \mathbb F_q^\times$ such that for all $i$,
\begin{equation} \label{eq: same character}
\chi(\bm{\epsilon}_i)=\epsilon_{i1} \dots \epsilon_{i\ell_i}=\epsilon.
\end{equation}

We now apply Theorem \ref{theorem: linear independence} to our setting of ACMPL's. We know that the hypothesis are verified:
\begin{itemize}
\item[(LW)] By the induction hypothesis, for any weight $w'<w$, the values $\mathfrak{Li}(\frak t;\bm{\epsilon})(\theta)$ with $(\frak t;\bm{\epsilon}) \in I$ and $w(\frak t)=w'$ are all $K$-linearly independent.

\item[(LD)] By \eqref{eq: non trivial relation MZV}, there exist $a \in A$ and $a_i \in A$ for $1 \leq i \leq n$ which are not all zero such that
\begin{equation*}
a+\sum_{i=1}^n a_i \mathfrak{Li}(\mathfrak{s}_i;\bm{\epsilon}_i)(\theta)=0.
\end{equation*}
\end{itemize}
Thus Theorem \ref{theorem: linear independence} implies that for all $(\mathfrak t;\bm{\epsilon}) \in I$, $f_{\mathfrak t,\bm{\epsilon}}(\theta)$ belongs to $K$ where $f_{\mathfrak t,\bm{\epsilon}}$ is given by
\begin{equation*}
f_{\mathfrak t;\bm{\epsilon}}:= \sum_i a_i(t) \mathfrak{Li}(s_{i(k+1)},\dots,s_{i \ell_i};\epsilon_{i(k+1)},\dots,\epsilon_{i \ell_i}).
\end{equation*}
Here, the sum runs through the set of indices $i$ such that $(\mathfrak t;\bm{\epsilon})=(s_{i1},\dots,s_{i k};\epsilon_{i1},\dots,\epsilon_{i k})$ for some $0 \leq k \leq \ell_i-1$.

We derive a direct consequence of the previous rationality result. Let $(\mathfrak t;\bm{\epsilon}) \in I$ and $\mathfrak t \neq \emptyset$. Then $(\mathfrak t;\bm{\epsilon})=(s_{i1},\dots,s_{i k};\epsilon_{i1},\dots,\epsilon_{i k})$ for some $i$ and $1 \leq k \leq \ell_i-1$. We denote by $J(\mathfrak t;\bm{\epsilon})$ the set of all such $i$. We know that there exists $b \in K$ such that
\begin{equation*}
b+f_{\mathfrak t;\bm{\epsilon}}=0,
\end{equation*}
or equivalently,
\begin{equation*}
b+\sum_{i \in J(\frak t;\bm{\epsilon})} a_i(t) \mathfrak{Li}(s_{i(k+1)},\dots,s_{i \ell_i};\epsilon_{i(k+1)},\dots,\epsilon_{i \ell_i})=0.
\end{equation*}
The ACMPL's appearing in the above equality belong to $\mathcal{AS}_{w-w(\mathfrak t)}$. By the induction hypothesis, we can suppose that $\epsilon_{i(k+1)}=\dots=\epsilon_{i \ell_i}=1$. Further, if $q-1 \nmid w-w(\frak t)$, then $a_i(t)=0$ for all $i \in J(\frak t;\bm{\epsilon})$. Therefore,  letting $\mathfrak{s}_i=(s_{i1},\dots,s_{i \ell_i};\epsilon_{i1},\dots,\epsilon_{i \ell_i})$ we can suppose that $s_{i2},\dots,s_{i \ell_i}$ are all divisible by $q-1$ and $\epsilon_{i2}=\dots=\epsilon_{i \ell_i}=1$. In particular, for all $i$, $\epsilon_{i1}=\chi(\bm{\epsilon}_i)=\epsilon$.

Now we want to solve \eqref{eq:equation for delta} and we can assume that $b=1$. We define
	\[ J:=I \cup \{(\mathfrak{s}_i;\bm{\epsilon}_i)\}\]
For $(\mathfrak t;\bm{\epsilon}) \in J$ we denote by $J_0(\frak t;\bm{\epsilon})$ consisting of $(\frak t';\bm{\epsilon}') \in I$ such that there exist $i$ and $0 \leq j <\ell_i$ so that $(\frak t;\bm{\epsilon})=(s_{i1},s_{i2},\dots,s_{ij};\epsilon,1,\dots,1)$ and $\frak t'=(s_{i1},s_{i2},\dots,s_{i(j+1)};\epsilon,1,\dots,1)$. In particular, for $(\frak t;\bm{\epsilon})=(\mathfrak{s}_i;\bm{\epsilon}_i)$, $J_0(\frak t;\bm{\epsilon})$ is the empty set. For $(\mathfrak t;\bm{\epsilon}) \in J \setminus \emptyset$, we also put 
	\[ m_{\frak t}:=\frac{w-w(\frak t)}{q-1} \in \mathbb Z^{\geq 0}. \]
Then it is clear that \eqref{eq:equation for delta} is equivalent finding $(\delta_{\mathfrak t,\bm{\epsilon}})_{(\mathfrak t;\bm{\epsilon}) \in J} \in \Mat_{1 \times |J|}(\overline K[t])$ such that
\begin{equation} \label{eq:delta}
\delta_{\mathfrak t,\bm{\epsilon}}=\delta_{\mathfrak t,\bm{\epsilon}}^{(-1)} (t-\theta)^{w-w(\frak t)}+\sum_{(\mathfrak t',\bm{\epsilon}') \in J_0(\frak t,\bm{\epsilon})} \delta_{\mathfrak t',\bm{\epsilon}'}^{(-1)} (t-\theta)^{w-w(\frak t)}, \quad \text{for all } (\frak t,\bm{\epsilon}) \in J \setminus \emptyset,
\end{equation}
and 
\begin{equation} \label{eq:delta emptyset}
\delta_{\mathfrak t,\bm{\epsilon}}=\delta_{\mathfrak t,\bm{\epsilon}}^{(-1)} (t-\theta)^{w-w(\frak t)}+\sum_{(\mathfrak t',\bm{\epsilon}') \in J_0(\frak t,\bm{\epsilon})} \delta_{\mathfrak t',\bm{\epsilon}'}^{(-1)} \gamma^{(-1)} (t-\theta)^{w-w(\frak t)}, \quad \text{for } (\frak t,\bm{\epsilon})=\emptyset.
\end{equation}
Here $\gamma^{q-1}=\epsilon$. In fact, for $(\frak t,\bm{\epsilon})=(\mathfrak{s}_i,\bm{\epsilon}_i)$, the corresponding equation becomes $\delta_{\mathfrak{s}_i,\bm{\epsilon}_i}=\delta_{\mathfrak{s}_i,\bm{\epsilon}_i}^{(-1)}$. Thus $\delta_{\mathfrak{s}_i,\bm{\epsilon}_i}=a_i(t) \in \mathbb F_q[t]$.

Letting $y$ be a variable, we denote by $v_y$ the valuation associated to the place $y$ of the field $\mathbb F_q(y)$. We put 
	\[ T:=t-t^q, \quad X:=t^q-\theta^q. \]

We claim that 
\begin{itemize}
\item[1)] For all $(\mathfrak t,\bm{\epsilon}) \in J \setminus \emptyset$, the polynomial $\delta_{\frak t,\bm{\epsilon}}$ is of the form
	\[ \delta_{\frak t,\bm{\epsilon}}=f_{\frak t} \left(X^{m_{\frak t}}+\sum_{i=0}^{m_{\frak t}-1} P_{\frak t,i}(T) X^i \right)\]
where 
\begin{itemize}
\item $f_{\frak t} \in \mathbb F_q[t]$,
\item for all $0 \leq i \leq m_{\frak t}-1$, $P_{\frak t,i}(y)$ belongs to $\mathbb F_q(y)$ with $v_y(P_{\frak t,i}) \geq 1$. 
\end{itemize}

\item[2)] For all $\mathfrak t \in J \setminus \emptyset$ and all $\frak t' \in J_0(\frak t)$, there exists $P_{\frak t,\frak t'} \in \mathbb F_q(y)$ such that
	\[ f_{\frak t'}=f_{\frak t} P_{\frak t,\frak t'}(T). \]
In particular, if $f_{\frak t}=0$, then $f_{\frak t'}=0$.
\end{itemize}

The proof is by induction on $m_{\frak t}$. We start with $m_{\frak t}=0$. Then $\frak t=\mathfrak{s}_i$ and $\bm{\epsilon}=\bm{\epsilon}_i$ for some $i$. We have observed that $\delta_{\mathfrak{s}_i,\bm{\epsilon}_i}=a_i(t) \in \mathbb F_q[t]$. Thus we are done. 

Suppose that the claim holds for all $(\frak t,\bm{\epsilon}) \in J \setminus \emptyset$ with $m_{\frak t}<m$. We now prove the claim for all $(\frak t,\bm{\epsilon}) \in J \setminus \emptyset$ with $m_{\frak t}=m$. In fact, we fix such $\frak t$ and want to find $\delta_{\frak t,\bm{\epsilon}} \in \overline K[t]$ such that
\begin{equation} \label{eq:delta1}
 \delta_{\mathfrak t,\bm{\epsilon}}=\delta_{\mathfrak t,\bm{\epsilon}}^{(-1)} (t-\theta)^{(q-1)m}+\sum_{(\mathfrak t',\bm{\epsilon}') \in J_0(\frak t;\bm{\epsilon})} \delta_{\mathfrak t',\bm{\epsilon}'}^{(-1)} (t-\theta)^{(q-1)m}.
\end{equation}
By the induction hypothesis, for all $(\frak t',\bm{\epsilon}') \in J_0(\frak t;\bm{\epsilon})$, we know that 
	\[ \delta_{\frak t',\bm{\epsilon}'}=f_{\frak t'} \left(X^{m_{\frak t'}}+\sum_{i=0}^{m_{\frak t'}-1} P_{\frak t',i}(T) X^i \right)\]
where 
\begin{itemize}
\item $f_{\frak t'} \in \mathbb F_q[t]$,
\item for all $0 \leq i \leq m_{\frak t'}-1$, $P_{\frak t',i}(y) \in \mathbb F_q(y)$ with $v_y(P_{\frak t,i}) \geq 1$. 
\end{itemize}
For $(\frak t',\bm{\epsilon}') \in J_0(\frak t;\bm{\epsilon})$, we write $\frak t'=(\frak t,(m-k)(q-1))$ with $0 \leq k < m$ and $k \not\equiv m \pmod{q}$, in particular $m_{\frak t'}=k$. We put $f_k=f_{\frak t'}$ and $P_{\frak t',i}=P_{k,i}$ so that  
\begin{equation} \label{eq:delta2}
\delta_{\frak t',\bm{\epsilon}'}=f_k \left(X^k+\sum_{i=0}^{k-1} P_{k,i}(T) X^i \right) \in \mathbb F_q[t,\theta^q].
\end{equation}

By Lemma \ref{lem: KuanLin}, $\delta_{\frak t,\bm{\epsilon}}$ belongs to $K[t]$, and 
$\deg_\theta \delta_{\frak t,\bm{\epsilon}} \leq mq$. Further, since $\delta_{\frak t,\bm{\epsilon}}$ is divisible by $(t-\theta)^{(q-1)m}$,  we write $\delta_{\frak t,\bm{\epsilon}}=F (t-\theta)^{(q-1)m}$ with $F \in K[t]$ and $\deg_\theta F \leq m$. Dividing \eqref{eq:delta1} by $(t-\theta)^{(q-1)m}$ and twisting once yields
\begin{equation} \label{eq:F}
F^{(1)}=F (t-\theta)^{(q-1)m}+\sum_{(\mathfrak t',\bm{\epsilon}') \in J_0(\frak t,\bm{\epsilon})} \delta_{\mathfrak t',\bm{\epsilon}'}.
\end{equation}
As $\delta_{\frak t',\bm{\epsilon}'} \in \mathbb F_q[t,\theta^q]$ for all $(\frak t',\bm{\epsilon}') \in J_0(\frak t;\bm{\epsilon})$, it follows that $F (t-\theta)^{(q-1)m} \in \mathbb F_q[t,\theta^q]$. As $\deg_\theta F \leq m$, we get
	\[ F=\sum_{0 \leq i \leq m/q} f_{m-iq} (t-\theta)^{m-iq}, \quad \text{for $f_{m-iq} \in \mathbb F_q[t]$}. \]
Thus
\begin{align*}
& F (t-\theta)^{(q-1)m}=\sum_{0 \leq i \leq m/q} f_{m-iq} (t-\theta)^{mq-iq}=\sum_{0 \leq i \leq m/q} f_{m-iq} X^{m-i}, \\
& F^{(1)}=\sum_{0 \leq i \leq m/q} f_{m-iq} (t-\theta^q)^{m-iq}=\sum_{0 \leq i \leq m/q} f_{m-iq} (T+X)^{m-iq}.
\end{align*}
Putting these and \eqref{eq:delta2} into \eqref{eq:F} gets
\begin{align*}
& \sum_{0 \leq i \leq m/q} f_{m-iq} (T+X)^{m-iq}\\
&=\sum_{0 \leq i \leq m/q} f_{m-iq} X^{m-i}+\sum_{\substack{0 \leq k<m \\ k \not\equiv m \pmod{q}}} f_k \left(X^k+\sum_{i=0}^{k-1} P_{k,i}(T) X^i \right).
\end{align*}
Comparing the coefficients of powers of $X$ yields the following linear system in the variables $f_0,\dots,f_{m-1}$:
\begin{equation*}
B_{\big| y=T} \begin{pmatrix}
f_{m-1} \\
\vdots \\
f_0
\end{pmatrix}=f_m \begin{pmatrix}
Q_{m-1} \\
\vdots \\
Q_0
\end{pmatrix}_{\big| y=T}.
\end{equation*}
Here for $0 \leq i \leq m-1$, $Q_i={m \choose i} y^{m-i} \in y\mathbb F_q[y]$ and $B=(B_{ij})_{0 \leq i,j \leq m-1} \in \Mat_m(\mathbb F_q(y))$ such that 
\begin{itemize}
\item $v_y(B_{ij}) \geq 1$ if $i>j$,
\item $v_y(B_{ij}) \geq 0$ if $i<j$,
\item $v_y(B_{ii})=0$ as $B_{ii}=\pm 1$.
\end{itemize}
The above properties follow from the fact that $P_{k,i} \in \mathbb F_q(y)$ and $v_y(P_{k,i}) \geq 1$.
Thus $v_y(\det B)=0$ so that $\det B \neq 0$. It follows that for all $0 \leq i \leq m-1$, $f_i=f_m P_i(T)$ with $P_i \in \mathbb F_q(y)$ and $v_y(P_i) \geq 1$ and we are done.

To conclude, we have to solve \eqref{eq:delta} for $(\frak t;\bm{\epsilon})=\emptyset$. We have some extra work as we have a factor $\gamma^{(-1)}$ on the right hand side of \eqref{eq:delta emptyset}. We use $\gamma^{(-1)}=\gamma/\epsilon$ and put $\delta:=\delta_{\emptyset,\emptyset}/\gamma \in \overline K[t]$. Then we have to solve 
\begin{equation} \label{eq:delta emptyset2}
\epsilon \delta=\delta^{(-1)} (t-\theta)^w+\sum_{(\mathfrak t',\bm{\epsilon}') \in J_0(\emptyset)} \delta_{\mathfrak t',\bm{\epsilon}'}^{(-1)} (t-\theta)^w.
\end{equation}

We distinguish two cases.
\subsubsection{Case 1: $q-1 \nmid w$, says $w=m(q-1)+r$ with $0<r<q-1$} ${}$\par

We know that for all $(\frak t',\bm{\epsilon}') \in J_0(\emptyset)$, says $\frak t'=((m-k)(q-1)+r)$ with $0 \leq k \leq m$  and $k \not\equiv m-r \pmod{q}$, 
\begin{equation} \label{eq:delta4}
\delta_{\frak t',\bm{\epsilon}'}=f_k \left(X^k+\sum_{i=0}^{k-1} P_{k,i}(T) X^i \right) \in \mathbb F_q[t,\theta^q]
\end{equation}
where 
\begin{itemize}
\item $f_k \in \mathbb F_q[t]$,
\item for all $0 \leq i \leq k-1$, $P_{k,i}(y)$ belongs to $\mathbb F_q(y)$ with $v_y(P_{k,i}) \geq 1$. 
\end{itemize}

By Lemma \ref{lem: KuanLin}, $\delta$ belongs to $K[t]$. We claim that $\deg_\theta \delta \leq mq$. Otherwise, we have $\deg_\theta \delta_\emptyset > mq$. Twisting \eqref{eq:delta emptyset2} once gets
\begin{equation*} 
\epsilon \delta^{(1)}=\delta (t-\theta^q)^w+\sum_{(\mathfrak t';\bm{\epsilon}') \in J_0(\emptyset)} \delta_{\mathfrak t',\bm{\epsilon}'} (t-\theta^q)^w.
\end{equation*}
As $\deg_\theta \delta > mq$, we compare the degrees of $\theta$ on both sides and obtain
	\[ q\deg_\theta \delta=\deg_\theta \delta+wq. \]
Thus $q-1 \mid w$, which is a contradiction. We conclude that $\deg_\theta \delta \leq mq$.

From \eqref{eq:delta emptyset2} we see that $\delta$ is divisible by $(t-\theta)^w$. Thus we write $\delta=F (t-\theta)^w$ with $F \in K[t]$ and $\deg_\theta F \leq mq-w=m-r$. Dividing \eqref{eq:delta emptyset2} by $(t-\theta)^w$ and twisting once yields
\begin{equation} \label{eq:F1}
\epsilon F^{(1)}=F (t-\theta)^w+\sum_{(\mathfrak t';\bm{\epsilon}') \in J_0(\emptyset)} \delta_{\mathfrak t'}.
\end{equation}
Since $\delta_{\frak t';\bm{\epsilon}'} \in \mathbb F_q[t,\theta^q]$  for all $(\frak t';\bm{\epsilon}') \in J_0(\emptyset)$, it follows that $F (t-\theta)^w \in \mathbb F_q[t,\theta^q]$. As $\deg_\theta F \leq m-r$, we write
	\[ F=\sum_{0 \leq i \leq (m-r)/q} f_{m-r-iq} (t-\theta)^{m-r-iq}, \quad \text{for $f_{m-r-iq} \in \mathbb F_q[t]$}. \]
It follows that
\begin{align*}
& F (t-\theta)^w=\sum_{0 \leq i \leq (m-r)/q} f_{m-r-iq} (t-\theta)^{mq-iq}=\sum_{0 \leq i \leq (m-r)/q} f_{m-r-iq} X^{m-i}, \\
& F^{(1)}=\sum_{0 \leq i \leq (m-r)/q} f_{m-r-iq} (t-\theta^q)^{m-r-iq}=\sum_{0 \leq i \leq (m-r)/q} f_{m-r-iq} (T+X)^{m-r-iq}.
\end{align*}
Putting these and \eqref{eq:delta4} into \eqref{eq:F1} yields
\begin{align*}
& \epsilon \sum_{0 \leq i \leq (m-r)/q} f_{m-r-iq} (T+X)^{m-r-iq} \\
&=\sum_{0 \leq i \leq (m-r)/q} f_{m-r-iq} X^{m-i}+\sum_{\substack{0 \leq k \leq m \\ k \not\equiv m-r \pmod{q}}} f_k \left(X^k+\sum_{i=0}^{k-1} P_{k,i}(T) X^i \right).
\end{align*}
Comparing the coefficients of powers of $X$ yields the following linear system in the variables $f_0,\dots,f_m$:
\begin{equation*}
B_{\big| y=T} \begin{pmatrix}
f_m \\
\vdots \\
f_0
\end{pmatrix}=0.
\end{equation*}
Here $B=(B_{ij})_{0 \leq i,j \leq m} \in \Mat_{m+1}(\mathbb F_q(y))$ such that 
\begin{itemize}
\item $v_y(B_{ij}) \geq 1$ if $i>j$,
\item $v_y(B_{ij}) \geq 0$ if $i<j$,
\item $v_y(B_{ii})=0$ as $B_{ii} \in \mathbb F_q^\times$.
\end{itemize}
The above properties follow from the fact that $P_{k,i} \in \mathbb F_q(y)$ and $v_y(P_{k,i}) \geq 1$.
Thus $v_y(\det B)=0$. Hence $f_0=\dots=f_m=0$. It follows that $\delta_{\emptyset;\emptyset}=0$ as $\delta=0$ and $\delta_{\frak t',\bm{\epsilon}'}=0$ for all $(\frak t';\bm{\epsilon}') \in J_0(\emptyset)$. We conclude that $\delta_{\frak t,\bm{\epsilon}}=0$ for all $(\frak t,\bm{\epsilon}) \in J$. In particular, for all $i$, $a_i(t)=\delta_{\mathfrak{s}_i,\bm{\epsilon}_i}=0$, which is a contradiction. Thus this case can never happen.

\subsubsection{Case 2: $q-1 \mid w$, says $w=m(q-1)$} ${}$\par

By similar arguments as above, we show that $\delta=F (t-\theta)^{(q-1)m}$ with $F \in K[t]$ of the form
	\[ F=\sum_{0 \leq i \leq m/q} f_{m-iq} (t-\theta)^{m-iq}, \quad \text{for $f_{m-iq} \in \mathbb F_q[t]$}. \]
Thus
\begin{align*}
& F (t-\theta)^{(q-1)m}=\sum_{0 \leq i \leq m/q} f_{m-iq} (t-\theta)^{mq-iq}=\sum_{0 \leq i \leq m/q} f_{m-iq} X^{m-i}, \\
& F^{(1)}=\sum_{0 \leq i \leq m/q} f_{m-iq} (t-\theta^q)^{m-iq}=\sum_{0 \leq i \leq m/q} f_{m-iq} (T+X)^{m-iq}.
\end{align*}
Putting these and \eqref{eq:delta2} into \eqref{eq:delta emptyset2} gets
\begin{align*}
& \epsilon \sum_{0 \leq i \leq m/q} f_{m-iq} (T+X)^{m-iq}\\
&=\sum_{0 \leq i \leq m/q} f_{m-iq} X^{m-i}+\sum_{\substack{0 \leq k<m \\ k \not\equiv m \pmod{q}}} f_k \left(X^k+\sum_{i=0}^{k-1} P_{k,i}(T) X^i \right).
\end{align*}

Comparing the coefficients of powers of $X$ yields 
	\[ \epsilon f_m=f_m \] 
and the following linear system in the variables $f_0,\dots,f_{m-1}$:
\begin{equation*}
B_{\big| y=T} \begin{pmatrix}
f_{m-1} \\
\vdots \\
f_0
\end{pmatrix}=f_m \begin{pmatrix}
Q_{m-1} \\
\vdots \\
Q_0
\end{pmatrix}_{\big| y=T}.
\end{equation*}
Here for $0 \leq i \leq m-1$, $Q_i={m \choose i} y^{m-i} \in y\mathbb F_q[y]$ and $B=(B_{ij})_{0 \leq i,j \leq m-1} \in \Mat_m(\mathbb F_q(y))$ such that 
\begin{itemize}
\item $v_y(B_{ij}) \geq 1$ if $i>j$,
\item $v_y(B_{ij}) \geq 0$ if $i<j$,
\item $v_y(B_{ii})=0$ as $B_{ii} \in \mathbb F_q^\times$.
\end{itemize}
The above properties follow from the fact that $P_{k,i} \in \mathbb F_q(y)$ and $v_y(P_{k,i}) \geq 1$.
Thus $v_y(\det B)=0$ so that $\det B \neq 0$. 

We distinguish two subcases.

\medskip
\noindent {\bf Subcase 1: $\epsilon \neq 1$.} ${}$\par
It follows that $f_m=0$. Then $f_0=\dots=f_{m-1}=0$. Thus $\delta_{\frak t,\bm{\epsilon}}=0$ for all $(\frak t,\bm{\epsilon}) \in J$. In particular, for all $i$, $a_i(t)=\delta_{\mathfrak{s}_i,\bm{\epsilon}_i}=0$. This is a contradiction and we conclude that this case can never happen.

\medskip
\noindent {\bf Subcase 2: $\epsilon=1$.} ${}$\par

It follows that $\gamma \in \mathbb F_q^\times$ and thus
\begin{itemize}
\item[1)] The polynomial $\delta_{\emptyset;\emptyset}=\delta \gamma$ is of the form
	\[ \delta_{\emptyset;\emptyset}=f_\emptyset \left(X^m+\sum_{i=0}^{m-1} P_{\emptyset,i}(T) X^i \right)\]
with
\begin{itemize}
\item $f_\emptyset \in \mathbb F_q[t]$,
\item for all $0 \leq i \leq m-1$, $P_{\emptyset,i}(y) \in  \mathbb F_q(y)$ with $v_y(P_{\emptyset,i}) \geq 1$. 
\end{itemize}

\item[2)] For all $(\frak t',\bm{\epsilon}') \in J_0(\emptyset)$, there exists $P_{\emptyset,\frak t'} \in \mathbb F_q(y)$ such that
	\[ f_{\frak t'}=f_\emptyset P_{\emptyset,\frak t'}(T). \]
\end{itemize}
Hence there exists a unique solution $(\delta_{\mathfrak t,\bm{\epsilon}})_{(\mathfrak t,\bm{\epsilon}) \in J} \in \Mat_{1 \times |J|}(K[t])$ of \eqref{eq:delta} up to a factor in $\mathbb F_q(t)$. Recall that for all $i$, $a_i(t)=\delta_{\mathfrak{s}_i,\bm{\epsilon}_i}$. Therefore, up to a scalar in $K^\times$, there exists at most one non trivial relation 
	\[ a \widetilde \pi^w+\sum_i a_i \Li \begin{pmatrix}
 \bm{\varepsilon}_i  \\
\mathfrak{s}_i  \end{pmatrix}=0 \]
with $a_i \in K$ and $\Li \begin{pmatrix}
 \bm{\varepsilon}_i  \\
\mathfrak{s}_i  \end{pmatrix} \in \mathcal{AS}_w$. Further, we must have $\bm{\varepsilon}_i=(1,\dots,1)$ for all $i$.

To conclude, it suffices to exhibit such a relation with $a \neq 0$. In fact, we recall $w=(q-1)m$ and then express $\Li(q-1)^m=\Li \begin{pmatrix}
 1 \\
q-1  \end{pmatrix}^m$ as a $K$-linear combination of ACMPL's of weight $w$.  By Theorem \ref{prop: weak Brown}, we can write
	\[ \Li(q-1)^m=\Li \begin{pmatrix}
 1 \\
q-1  \end{pmatrix}^m=\sum_i a_i \Li \begin{pmatrix}
 \bm{\varepsilon}_i  \\
\mathfrak{s}_i  \end{pmatrix}=0, \quad \text{where $a_i \in K$, $\Li \begin{pmatrix}
 \bm{\varepsilon}_i  \\
\mathfrak{s}_i  \end{pmatrix} \in \mathcal{AS}_w.$} \] 
We note that $\Li(q-1)=\zeta_A(q-1)=-D^{-1}_1 \widetilde \pi^{q-1}$. Thus
	\[ (-D_1)^m \widetilde \pi^w-\sum_i a_i \Li \begin{pmatrix}
 \bm{\varepsilon}_i  \\
\mathfrak{s}_i  \end{pmatrix}=0, \]
which is the desired relation. Hence we are done.


\section{Applications on AMZV's and Zagier-Hoffman's conjectures in positive characteristic} \label{sec:applications}

In this section we give two applications of the study of ACMPL's. 

First we apply Theorem \ref{thm:ACMPL} to prove Theorem \ref{thm: ZagierHoffman AMZV} which calculates the dimensions of the vector space $\mathcal{AZ}_w$ of alternating multiple zeta values in positive characteristic (AMZV's) of fixed weight introduced by Harada \cite{Har21}. Consequently we determine all linear relations for AMZV's. To do so we develop an algebraic theory to obtain a weak version of Brown's theorem for AMZV's. Then we deduce that $\mathcal{AZ}_w$ and $\mathcal{AL}_w$ are the same and conclude. Contrary to the setting of MZV's, although the results are neat, we are unable to obtain neither sharp upper bounds nor sharp lower bounds for $\mathcal{AZ}_w$ for general $w$ without the theory of ACMPL's.

Second we restrict our attention to MZV's and determine all linear relations among MZV's. In particular, we obtain a proof of Zagier-Hoffman's conjectures in positive characteristic in full generality (i.e., Theorem \ref{thm: ZagierHoffman}) and generalize the work of one of the authors \cite{ND21}.

\subsection{Linear relations among AMZV's} \label{sec: application AMZV}

\subsubsection{Preliminaries}

For $d \in \mathbb{Z}$ and for $\mathfrak{s}=(s_1,\dots,s_n) \in \ensuremath \mathbb{N}^n$, recalling $S_d(\mathfrak{s})$ and $S_{<d}(\mathfrak{s})$ given in \S1.1.2,
and further letting $\begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  =  \begin{pmatrix}
 \varepsilon_1 & \dots & \varepsilon_n \\
s_1 & \dots & s_n \end{pmatrix} $ be a positive array, we recall (see \S1.1.2)
\begin{equation*}
S_d \begin{pmatrix}
\bm{\varepsilon} \\ \mathfrak{s}
\end{pmatrix}  = \sum\limits_{\substack{a_1, \dots, a_n \in A_{+} \\ d = \deg a_1> \dots > \deg a_n\geq 0}} \dfrac{\varepsilon_1^{\deg a_1} \dots \varepsilon_n^{\deg a_n }}{a_1^{s_1} \dots a_n^{s_n}} \in K
\end{equation*}
and 
\begin{equation*}
    S_{<d} \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  = \sum\limits_{\substack{a_1, \dots, a_n \in A_{+} \\ d > \deg a_1> \dots > \deg a_n\geq 0}} \dfrac{\varepsilon_1^{\deg a_1} \dots \varepsilon_n^{\deg a_n }}{a_1^{s_1} \dots a_n^{s_n}} \in K.
\end{equation*}
One verifies easily the following formulas:
\begin{align*}
    S_{d} \begin{pmatrix}
 \varepsilon \\
s  \end{pmatrix}  &= \varepsilon^d S_d(s),\\
S_d \begin{pmatrix}
 1& \dots & 1 \\
s_1 & \dots & s_n \end{pmatrix}  &= S_{d}(s_1, \dots, s_n),\\
S_{<d} \begin{pmatrix}
 1& \dots & 1 \\
s_1 & \dots & s_n \end{pmatrix}  &= S_{<d}(s_1, \dots, s_n),\\
S_{d} \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  &= S_{d} \begin{pmatrix}
 \varepsilon_1  \\
s_1  \end{pmatrix} S_{<d} \begin{pmatrix}
 \bm{\varepsilon}_{-}  \\
\mathfrak{s}_{-}  \end{pmatrix}.
\end{align*}

Harada \cite{Har21} introduced the alternating multiple zeta value (AMZV) as follows;
\begin{equation*}
    \zeta_A \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  = \sum \limits_{d \geq 0} S_d \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  = \sum\limits_{\substack{a_1, \dots, a_n \in A_{+} \\ \deg a_1> \dots > \deg a_n\geq 0}} \dfrac{\varepsilon_1^{\deg a_1} \dots \varepsilon_n^{\deg a_n }}{a_1^{s_1} \dots a_n^{s_n}}  \in K_{\infty}.
\end{equation*}

Using Chen's formula (see \cite{Che15}), Harada proved that for $s, t \in \mathbb{N}$ and $\varepsilon, \epsilon \in \mathbb{F}_q^{\times}$, we have 
\begin{equation*}
S_d \begin{pmatrix}
 \varepsilon \\
s  \end{pmatrix} S_d \begin{pmatrix}
 \epsilon  \\
t  \end{pmatrix}  = S_d \begin{pmatrix}
 \varepsilon\epsilon  \\
s+t  \end{pmatrix}  + \sum \limits_i \Delta^i_{s,t} S_d \begin{pmatrix}
 \varepsilon\epsilon  & 1 \\
s+t-i & i \end{pmatrix} ,
\end{equation*}
where 
\begin{equation} \label{eq:Delta Chen}
    \Delta^i_{s,t} = \begin{cases}
			(-1)^{s-1} {i - 1  \choose s - 1} +  (-1)^{t-1} {i-1 \choose t-1} & \quad  \text{if } q - 1 \mid i \text{ and } 0 < i < s + t, \\
            0 & \quad \text{otherwise.}
		 \end{cases}
\end{equation}

He then proved similar results for products of AMZV's (see \cite{Har21}):
\begin{proposition} \label{sums}
Let $ \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} $, $ \begin{pmatrix}
 \bm{\epsilon}  \\
\mathfrak{t}  \end{pmatrix} $ be two positive arrays. Then 
\begin{enumerate}
    \item There exist $f_i \in \mathbb{F}_p$ and arrays $ \begin{pmatrix}
 \bm{\mu}_i  \\
\mathfrak{u}_i  \end{pmatrix} $ with $ \begin{pmatrix}
 \bm{\mu}_i  \\
\mathfrak{u}_i  \end{pmatrix}  \leq  \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  +  \begin{pmatrix}
 \bm{\epsilon}  \\
\mathfrak{t}  \end{pmatrix}  $ and $\depth(\mathfrak{u}_i) \leq \depth(\mathfrak{s}) + \depth(\mathfrak{t})$ for all $i$  such that 
    \begin{equation*}
        S_d \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} S_d \begin{pmatrix}
 \bm{\epsilon}  \\
\mathfrak{t}  \end{pmatrix}  = \sum \limits_i f_i S_d \begin{pmatrix}
 \bm{\mu}_i  \\
\mathfrak{u}_i  \end{pmatrix}  \quad \text{for all } d \in \mathbb{Z}.
    \end{equation*}
    \item There exist $f'_i \in \mathbb{F}_p$ and arrays $ \begin{pmatrix}
 \bm{\mu}'_i  \\
\mathfrak{u}'_i  \end{pmatrix} $ with $ \begin{pmatrix}
 \bm{\mu}'_i  \\
\mathfrak{u}'_i  \end{pmatrix}  \leq  \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  +  \begin{pmatrix}
 \bm{\epsilon}  \\
\mathfrak{t}  \end{pmatrix}  $ and $\depth(\mathfrak{u}'_i) \leq \depth(\mathfrak{s}) + \depth(\mathfrak{t})$ for all $i$  such that 
    \begin{equation*}
        S_{<d} \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} S_{<d} \begin{pmatrix}
 \bm{\epsilon}  \\
\mathfrak{t}  \end{pmatrix}  = \sum \limits_i f'_i S_{<d} \begin{pmatrix}
 \bm{\mu}'_i  \\
\mathfrak{u}'_i  \end{pmatrix}  \quad \text{for all } d \in \mathbb{Z}.
    \end{equation*}
        \item There exist $f''_i \in \mathbb{F}_p$ and arrays $ \begin{pmatrix}
 \bm{\mu}''_i  \\
\mathfrak{u}''_i  \end{pmatrix} $ with $ \begin{pmatrix}
 \bm{\mu}''_i  \\
\mathfrak{u}''_i  \end{pmatrix}  \leq  \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  +  \begin{pmatrix}
 \bm{\epsilon}  \\
\mathfrak{t}  \end{pmatrix}  $ and $\depth(\mathfrak{u}''_i) \leq \depth(\mathfrak{s}) + \depth(\mathfrak{t})$ for all $i$  such that 
    \begin{equation*}
        S_d \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} S_{<d} \begin{pmatrix}
 \bm{\epsilon}  \\
\mathfrak{t}  \end{pmatrix}  = \sum \limits_i f''_i S_d \begin{pmatrix}
 \bm{\mu}''_i  \\
\mathfrak{u}''_i  \end{pmatrix}  \quad \text{for all } d \in \mathbb{Z}.
    \end{equation*}
\end{enumerate}
\end{proposition}

We denote by $\mathcal{AZ}$ the $K$-vector space generated by the AMZV's and $\mathcal{AZ}_w$ the $K$-vector space generated by the AMZV's of weight $w$. It follows from Proposition~\ref{sums} that $\mathcal{AZ}$ is a $K$-algebra under the multiplication of $K_{\infty}$.

\subsubsection{Proof of Theorem \ref{thm: ZagierHoffman AMZV}} \label{sec:proof Theorem A}

We can extend an algebraic theory for AMZV's which follow the same line as that in \S \ref{sec:algebraic part}. In particular, we can for Todd's operations $\mathcal B^*_{\Sigma,V}$ and $\mathcal C_{\Sigma,V}$ for the setting of AMZV's. 

Consequently, we obtain a weak version of Brown's theorem for AMZV's as follows.
\begin{proposition} \label{prop: weak Brown AMZV}
The set of all elements $\zeta_A \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}$ such that $\zeta_A \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  \in \mathcal{AT}_w$ forms a set of generators for $\mathcal{AZ}_w$. Here we recall that $\mathcal{AT}_w$ is the set of all AMZV's $\zeta_A \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}  =  \Li \begin{pmatrix}
 \varepsilon_1 & \dots & \varepsilon_n \\
s_1 & \dots & s_n \end{pmatrix} $ of weight $w$ such that $s_1, \dots, s_{n-1} \leq q$ and $s_n < q$ introduced in the paragraph preceding Proposition \ref{prop: weak Brown}. 
\end{proposition}

As a direct consequence, we get
\begin{theorem} \label{thm:bridge}
The $K$-vector space $\mathcal{AZ}_w$ of AMZV's of weight $w$ and the $K$-vector space $\mathcal{AL}_w$ of ACMPL's of weight $w$ are the same.
\end{theorem}

By this identification we apply Theorem \ref{thm:ACMPL} to obtain Theorem \ref{thm: ZagierHoffman AMZV}.

\subsubsection{Sharp bounds without ACMPL's} \label{sec:without ACMPL}

To end this section we mention that without the theory of ACMPL's it seems very hard to obtain for arbitrary weight $w$
\begin{itemize}
\item either the sharp upper bound $\dim_K \mathcal{AZ}_w \leq s(w)$,
\item or the sharp lower bound $\dim_K \mathcal{AZ}_w \geq s(w)$.
\end{itemize}

We are only able to do so for small weights by ad hoc arguments. We collect the results below, sketch some ideas of the proofs and omit long computations.

\begin{proposition} \label{prop:Brown adhoc}
Let $w \leq 2q-2$. Then $\dim_K \mathcal{AZ}_w \leq s(w)$.
\end{proposition} 

\begin{proof}
We denote by $\mathcal{AT}_w^1$ the subset of AMZV's $\zeta_A \begin{pmatrix}
	\bm{\epsilon} \\ \mathfrak{s}
	\end{pmatrix}$ of $\mathcal{AT}_w$ such that $\epsilon_i=1$ whenever $s_i=q$ (i.e., the character corresponding to $q$ is always 1) and by $\langle \mathcal{AT}_w^1 \rangle$ the $K$-vector space spanned by the AMZV's in $\mathcal{AT}_w^1$. We see that $|\mathcal{AT}_w^1|=s(w)$. Thus it suffices to prove that $\langle \mathcal{AT}_w^1 \rangle=\mathcal{AZ}_w$.

Recall that for any $\epsilon \in \mathbb F_q^\times$, we recall the relation
\begin{equation*}
R_{\varepsilon} \colon \quad  S_d \begin{pmatrix}
 \varepsilon\\
q  \end{pmatrix}  + \varepsilon^{-1}D_1 S_{d+1} \begin{pmatrix}
 \varepsilon& 1 \\
1 & q-1  \end{pmatrix}  =0, 
\end{equation*}
where $D_1 = \theta^q - \theta$. 

We recall that the coefficients $\Delta^i_{s,t}$ are given in \eqref{eq:Delta Chen}. Let $U=(u_1,\dots,u_n)$ and $W=(w_1,\dots,w_r)$ be tuples of positive integers such that $u_n \leq q-1$ and $w_1,\dots,w_r \leq q$. Let $\bm{\epsilon}=(\epsilon_1,\dots,\epsilon_n)\in(\mathbb F_q^\times)^n$ and $\bm{\lambda}=(\lambda_1,\dots,\lambda_r)\in (\mathbb F_q^\times)^r$. By direct calculations, we show that $\mathcal B_{\bm{\epsilon},U} \mathcal C_{\bm{\lambda},W} (R_{\epsilon})$ can be written as
\begin{align*}
&S_d\begin{pmatrix}
\bm{\epsilon} & \epsilon & \bm{\lambda}\\
U & q & W
\end{pmatrix}+S_d\begin{pmatrix}
\bm{\epsilon} & \epsilon \lambda_1 & \bm{\lambda}_-\\
U & q + w_1 & W_-
\end{pmatrix}\\
& +\Delta^{q-1}_{q,w_1}S_d\begin{pmatrix}
\bm{\epsilon} & \epsilon \lambda_1 & 1 & \bm{\lambda}_-\\
U & w_1 + 1 & q - 1 & W_-
\end{pmatrix}\\ &+\Delta^{q-1}_{q,w_1}S_d\begin{pmatrix}
\bm{\epsilon} & \epsilon \lambda_1 & \lambda_2 &  \lambda_3 & \dots & \lambda_r\\
U & w_1 + 1 & q + w_2 - 1 & w_3 & \dots & w_r
\end{pmatrix}\\
& +\Delta^{q-1}_{q,w_1} \sum_{i=2}^{r-1}\prod_{j=2}^i(\Delta_{q-1,w_j}^{q-1}+1)S_d\begin{pmatrix}
\bm{\epsilon} & \epsilon \lambda_1 & \lambda_2 \dots \lambda_i & \lambda_{i+1} &  \lambda_{i+2}  \dots  \lambda_r\\
U & w_1 + 1 & w_2  \dots  w_i& q + w_{i+1} - 1 & w_{i+2}  \dots  w_r
\end{pmatrix}\\
& +\Delta^{q-1}_{q,w_1} \sum_{i=2}^{r}\prod_{j=2}^i(\Delta^{q-1}_{q-1,w_j}+1)S_d \begin{pmatrix}
\bm{\epsilon} & \epsilon \lambda_1 & \lambda_2 & \dots & \lambda_i & 1 &  \lambda_{i+1} & \dots & \lambda_r\\
U & w_1 + 1 & w_2 & \dots & w_i& q - 1 & w_{i+1} & \dots & w_r
\end{pmatrix} \\
&+\epsilon^{-1}D_1S_d\begin{pmatrix}
\bm{\epsilon} & \epsilon & 1 & \bm{\lambda}\\ U & 1 & q - 1 & W
\end{pmatrix}+\epsilon^{-1}D_1S_d\begin{pmatrix}
\bm{\epsilon} & \epsilon & \lambda_1 & \bm{\lambda}_-\\ 
U & 1 & q + w_1 - 1 & W_-
\end{pmatrix}\\
&+\epsilon^{-1}D_1S_d\begin{pmatrix}
\epsilon_1 & \dots & \epsilon_{n-1} & \epsilon_n\epsilon& 1 & \bm{\lambda}\\
u_1 & \dots & u_{n-1} & u_n + 1 & q - 1 & W
\end{pmatrix}\\
&+\epsilon^{-1}D_1S_d\begin{pmatrix}
\epsilon_1 & \dots & \epsilon_{n-1} & \epsilon_n\epsilon& \lambda_1 & \bm{\lambda}_-\\
u_1 & \dots & u_{n-1} & u_n + 1 & q + w_1 - 1 & W_-
\end{pmatrix}\\
&+\epsilon^{-1}D_1\sum_{i=1}^{r-1}\prod_{j=1}^i(\Delta_{q-1,w_j}^{q-1}+1)S_d\begin{pmatrix}
\bm{\epsilon} & \epsilon & \lambda_1 & \dots & \lambda_i & \lambda_{i+1} & \lambda_{i+2} & \dots & \lambda_r\\
U & 1 & w_1 &\dots & w_i & q + w_{i+1} - 1 & w_{i+2} &\dots & w_r
\end{pmatrix}\\
&+\epsilon^{-1}D_1\sum_{i=1}^{r}\prod_{j=1}^i(\Delta^{q-1}_{q-1,w_j}+1)S_d
\begin{pmatrix}
\bm{\epsilon}&\epsilon&\lambda_1\dots\lambda_i&1&\lambda_{i+1}\dots\lambda_r\\
U&1&w_1\dots w_i&q-1&w_{i+1}\dots w_r
\end{pmatrix}\\
&+\epsilon^{-1}D_1\sum_{i=1}^{r}\prod_{j=1}^i(\Delta^{q-1}_{q-1,w_j}+1)S_d\begin{pmatrix}
\epsilon_1 \dots \epsilon_{n-1}&\epsilon_n\epsilon&\lambda_1 \dots \lambda_i&1&\lambda_{i+1} \dots \lambda_r\\
u_1 \dots u_{n-1}&u_n+1&w_1 \dots w_i&q-1&w_{i+1} \dots w_r
\end{pmatrix}\\%(u_1,\cdots,u_{n-1},u_n+1,w_1,\cdots,w_i,q-1,w_{i+1},\cdots,w_r;\epsilon_1,\cdots,\epsilon_{n-1},\epsilon_n\epsilon,1,\lambda_1,\cdots,\lambda_i,1,\lambda_{i+1},\cdots,\lambda_r)+\\
+&\epsilon^{-1}D_1\sum_{i=1}^{r-1}\prod_{j=1}^i(\Delta^{q-1}_{q-1,w_j}+1)S_d
\begin{pmatrix}
\epsilon_1  \dots  \epsilon_{n-1}&\epsilon_n\epsilon&\lambda_1 \dots \lambda_i& \lambda_{i+1}&\lambda_{i+2} \dots \lambda_r\\
u_1  \dots  u_{n-1} &u_n+1&w_1 \dots w_i& q+w_{i+1}-1&w_{i+2} \dots w_r
\end{pmatrix} \\
&=0.
\end{align*}
We denote by $(*)$ this formula.

We denote by $H_r$ the following claim: for any tuples
of positive integers $U$ and $W=(w_1,\dots,w_r)$ of depth $r$, $\bm{\epsilon} \in(\mathbb F_q^\times)^{\depth U}$ of any depth,
$\bm{\lambda}=(\lambda_1,\dots,\lambda_r)\in(\mathbb F_q^\times)^{r}$,  and $\epsilon \in \mathbb F_q^\times$ such that
$w(U)+w(W)+q=w$, the AMZV's $\zeta_A\begin{pmatrix}
\bm{\epsilon} & \epsilon & \bm{\lambda}\\
U & q & W
\end{pmatrix}$ and $\zeta_A\begin{pmatrix}
\bm{\epsilon} & \epsilon & \bm{\lambda}_-\\
U & q+w_1 & W_-
\end{pmatrix}$  belong to $\langle\mathcal{AT}_w^1\rangle$.

We claim that $H_r$ holds for any $r \geq 0$. The proof is by induction on $r$. For $r=0$, we know that $W=\emptyset$. Letting $U=(u_1,\dots,u_n)$ and $\bm{\epsilon}=(\epsilon_1,\dots,\epsilon_n)$ we apply the formula $(*)$  to get an explicit expression for $\mathcal
B_{\bm{\epsilon},U}(R_{\epsilon})$ given by
\begin{align*}
    S_d\begin{pmatrix}
    \bm{\epsilon} & \epsilon\\
    U & q
    \end{pmatrix}&+\epsilon^{-1}D_1S_d\begin{pmatrix}
    \bm{\epsilon} & \epsilon& 1\\
    U & 1 & q - 1
    \end{pmatrix}\\
    &+\epsilon^{-1}D_1S_d\begin{pmatrix}
   \epsilon_1 & \dots & \epsilon_{n-1} & \epsilon_n \epsilon & 1 \\
    u_1 & \dots & u_{n-1} & u_n + 1 & q - 1  
    \end{pmatrix}=0.
\end{align*}
Since $u_i \leq w(U)=w-q \leq q-2$, we deduce that $\zeta_A\begin{pmatrix}
\bm{\epsilon} & \epsilon\\
U & q
\end{pmatrix}\in\langle \mathcal{AT}_w^1 \rangle$ as required.

Suppose that $H_{r'}$ holds for any $r'<r$. We now show that $H_r$ holds. We proceed again by induction on $w_1$. For $w_1=1$, letting $U=(u_1,\dots,u_n)$ and $\bm{\epsilon}=(\epsilon_1,\dots,\epsilon_n)$ we apply the formula $(*)$ to get an explicit expression for $\mathcal B_{\bm{\epsilon},U} \mathcal C_{\bm{\lambda},W} (R_{\epsilon})$. As $w(U)+w(W)=w-q \leq q-2$, by induction we deduce that all the terms except the first two ones in this expression belong to $\langle \mathcal{AT}_w^1 \rangle$. Thus for any $\epsilon \in \mathbb F_q^\times$,
\begin{equation} \label{eq:alg_2q-2}
\zeta_A\begin{pmatrix}
\bm{\epsilon} & \epsilon & \bm{\lambda}\\
U & q & W
\end{pmatrix}+\zeta_A\begin{pmatrix}
\bm{\epsilon} & \epsilon \lambda_1 & \bm{\lambda}_-\\
U & q + 1 & W_-
\end{pmatrix} \in \langle \mathcal{AT}_w^1 \rangle.
\end{equation}
We take $\epsilon=1$. As the first term lies in $\mathcal{AT}_w^1$ by definition, we deduce that  
\begin{align*}
\zeta_A\begin{pmatrix}
\bm{\epsilon} & \lambda_1 & \bm{\lambda}_-\\
U & q + 1 & W_-
\end{pmatrix} \in \langle \mathcal{AT}_w^1 \rangle.
\end{align*}
Thus in \eqref{eq:alg_2q-2} we now know that the second term lies in $\langle \mathcal{AT}_w^1 \rangle$, which implies that 
	\[ \zeta_A\begin{pmatrix}
\bm{\epsilon} & \epsilon & \bm{\lambda}\\
U & q & W
\end{pmatrix} \in \langle \mathcal{AT}_w^1 \rangle. \]

We suppose that $H_r$ holds for all $W'=(w_1',\dots,w_r')$ such that $w_1'<w_1$. We have to show that $H_r$ holds for all $W=(w_1,\dots,w_r)$. The proof follows the same line as before. Letting $U=(u_1,\dots,u_n)$ and $\bm{\epsilon}=(\epsilon_1,\dots,\epsilon_n)$ we apply the formula $(*)$ to get an explicit expression for $\mathcal B_{\bm{\epsilon},U} \mathcal C_{\bm{\lambda},W} (R_{\epsilon})$. As $w(U)+w(W)=w-q \leq q-2$, by induction we deduce that all the terms except the first two ones in this expression belong to $\langle \mathcal{AT}_w^1 \rangle$. Thus for any $\epsilon \in \mathbb F_q^\times$,
\begin{equation} \label{eq:alg_2q-2 2}
\zeta_A\begin{pmatrix}
\bm{\epsilon} & \epsilon & \bm{\lambda}\\
U & q & W
\end{pmatrix}+\zeta_A\begin{pmatrix}
\bm{\epsilon} & \epsilon \lambda_1 & \bm{\lambda}_-\\
U & q + w_1 & W_-
\end{pmatrix} \in \langle \mathcal{AT}_w^1 \rangle.
\end{equation}
We take $\epsilon=1$ and deduce
\begin{align*}
\zeta_A\begin{pmatrix}
\bm{\epsilon} & \lambda_1 & \bm{\lambda}_-\\
U & q + w_1 & W_-
\end{pmatrix} \in \langle \mathcal{AT}_w^1 \rangle.
\end{align*}
Thus in \eqref{eq:alg_2q-2 2} the second term lies in $\langle \mathcal{AT}_w^1 \rangle$, which implies 
	\[ \zeta_A\begin{pmatrix}
\bm{\epsilon} & \epsilon & \bm{\lambda}\\
U & q & W
\end{pmatrix} \in \langle \mathcal{AT}_w^1 \rangle. \]
The proof is complete.
\end{proof}

\begin{remark} \label{rem:2q-1}
The condition $w \leq 2q-2$ is essential in the previous proof as it allows us to significantly simplify the expression of $\mathcal B_{\bm{\epsilon},U} \mathcal C_{\bm{\lambda},W} (R_{\epsilon})$ (see Eq. \eqref{eq:alg_2q-2 2}). For $w=2q-1$ the situation is already complicated but we can manage to prove Proposition \ref{prop:Brown adhoc}. Unfortunately, we are not be able to extend it to $w=2q$.
\end{remark}

\begin{proposition}
Let $w \leq 3q-2$. Then $\dim_K \mathcal{AZ}_w \geq s(w)$.
\end{proposition} 

\begin{proof}
We denote by $\mathcal{AT}_w'$ the subset of AMZV's as follows. 
\begin{itemize}
\item For $1 \leq w \leq 2q-2$, $\mathcal{AT}_w'$ consists of $\zeta_A \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix}$ of weight $w$ such that if we write $\mathfrak{s}=(s_1,\dots,s_r)$, then $q-1 \nmid s_i$. 

\item For $2q-1 \leq w \leq 3q-2$, we consider $I_w$ the set of tuples $\mathfrak{s}=(s_1,\dots,s_r)$ of weight $w$ where $s_i \neq q, 2q-1, 2q, 3q-2$ for all $i$ satisfying
\begin{itemize}
\item if $s_{i+1}=q$ or $2q-1$ for some $i=1,2,\ldots, r-1$, then $q-1 | s_i$,
\item $\mathfrak{s}_i\neq (q-1, q-1, q)$ when $w=3q-2$,
\end{itemize}
and define
	\[ \mathcal{AT}_w':=\left\{\zeta_A\begin{pmatrix}
	\bm{\epsilon} \\ \mathfrak{s}
	\end{pmatrix} : \mathfrak{s} \in I_w, \text{ and } \epsilon_i=1 \text{ whenever } s_i=q \right\}.\]
\end{itemize}
Then for $w \leq 3q-2$, one shows that 
	\[ |\mathcal{AT}_w'|=s(w). \]
Further, by construction, for any $(\mathfrak{s};\bm{\epsilon})=(s_1,\dots,s_r;\epsilon_1,\dots,\epsilon_r) \in \ensuremath \mathbb{N}^r \times (\mathbb F_q^\times)^r$, if $\zeta_A \begin{pmatrix}
 \bm{\varepsilon}  \\
\mathfrak{s}  \end{pmatrix} \in \mathcal{AT}_w'$, then $\zeta_A \begin{pmatrix}
s_1 & \dots & s_{r-1} \\
\epsilon_1 & \dots & \epsilon_{r-1}
\end{pmatrix}$ belongs to $\mathcal{AT}_{w-s_r}'$. This property allows us to apply Theorem~\ref{theorem: linear independence} and show by induction on $w \leq 3q-2$ that the AMZV's in $\mathcal{AT}_w'$ are all linearly independent over $K$. The proof is similar to that of Theorem~\ref{thm: trans ACMPL}. We apply Theorem \ref{theorem: linear independence} and reduce to solve a system of $\sigma$-linear equations. By direct but complicated calculations, we show that there does not exist non trivial solutions and we are done. 
\end{proof}

\begin{remark}
1) We note that the MZV's $\zeta_A(1,2q-2)$ and $\zeta_A(2q-1)$ (resp. $\zeta_A(1,3q-3)$ and $\zeta_A(3q-2)$) are linearly dependent over $K$. This explains the above ad hoc construction of $\mathcal{AT}_w'$.

\noindent 2) Despite extensive numerical experiments, we cannot find a suitable basis $\mathcal{AT}_w'$ for the case $w=3q-1$.
\end{remark}

Consequently, by combining the previous results we obtain
\begin{itemize}
\item for $w \leq 2q-1$ a basis $\mathcal{AT}_w^1$ of $\mathcal{AZ}_w$ consisting of AMZV's of weight $w$  (see Remark \ref{rem:2q-1});
\item for $w \leq 3q-2$ another basis $\mathcal{AT}_w'$ of $\mathcal{AZ}_w$ consisting of AMZV's of weight~$w$;
\item a direct proof of Theorem \ref{thm: ZagierHoffman AMZV} for $w \leq 2q-1$, which generalizes \cite[Theorem~D]{ND21}.
\end{itemize}

\subsection{Zagier-Hoffman's conjectures in positive characteristic} \label{sec: application MZV}

\subsubsection{Known results}

We use freely the notation introduced in \S \ref{sec:ZagierHoffman}. We recall that for $w \in \ensuremath \mathbb{N}$, $\mathcal Z_w$ denotes the $K$-vector space spanned by the MZV's of weight~$w$ and $\mathcal T_w$ denotes the set of $\zeta_A(\mathfrak{s})$ where $\mathfrak{s}=(s_1,\ldots,s_r) \in \ensuremath \mathbb{N}^r$ of weight $w$ with $1\leq s_i\leq q$ for $1\leq i\leq r-1$ and $s_r<q$.


Recall that the main results of \cite{ND21} states that 
\begin{itemize}
\item For all $w \in \ensuremath \mathbb{N}$ we always have $\dim_K \mathcal Z_w \leq d(w)$  (see \cite[Theorem A]{ND21}). 

\item For $w \leq 2q-2$ we have $\dim_K \mathcal Z_w \geq d(w)$  (see \cite[Theorem B]{ND21}). In particular, Conjecture \ref{conj: basis} holds for $w \leq 2q-2$ (see \cite[Theorem D]{ND21}). 
\end{itemize}
However, as stated in \cite[Remark 6.3]{ND21} it would be very difficult to extend the method of \cite{ND21} for general weights.

As an application of our main results, we present a proof of Theorem \ref{thm: ZagierHoffman} which settles both Conjectures \ref{conj: dimension} and \ref{conj: basis}.

\subsubsection{Proof of Theorem \ref{thm: ZagierHoffman}} \label{sec:proof Theorem B}
As we have already known the sharp upper bound for $\mathcal Z_w$ (see \cite[Theorem A]{ND21}), Theorem \ref{thm: ZagierHoffman} follows immediately from the following proposition.

\begin{proposition} \label{prop:lower bound}
For all $w \in \ensuremath \mathbb{N}$ we have $\dim_K \mathcal Z_w \geq d(w)$.
\end{proposition}

\begin{proof}
We denote by $\mathcal S_w$ the set of MZV's consisting of $\Li(s_1,\ldots,s_r)$ of weight $w$ with  $q \nmid s_i$ for all $i$. Then $\mathcal S_w$ is a subset of $\mathcal{AS}_w$ and belong to $\mathcal Z_w$ by Theorem \ref{thm:bridge}. Further, by \S \ref{sec:AS_w}, $|\mathcal S_w|=d(w)$. By Theorem \ref{thm: trans ACMPL} we deduce that elements in $\mathcal S_w$ are all linearly independent over $K$. Therefore, $\dim_K \mathcal Z_w \geq |\mathcal S_w|=d(w)$. 
\end{proof}


