$h^1$, $h_1$ of Anderson t-motives, systems of affine equations and non-commutative determinants

TL;DR

This paper introduces two affine equation systems for Anderson t-motives to compute cohomology groups, develops a non-commutative determinant concept, and provides explicit calculations for specific motives, advancing the algebraic understanding of t-motives.

Contribution

It defines new systems of affine equations for Anderson t-motives, introduces a non-commutative determinant, and performs explicit calculations for certain classes of motives.

Findings

Defined systems of affine equations for cohomology calculation.

Introduced a non-commutative determinant in the Anderson ring.

Calculated determinants for Drinfeld modules and t-motives of specific dimensions.

Abstract

The authors defined in "$h^1\ne h_1$ for Anderson t-motives" the notion of an affine equation associated to a t-motive $M$. Here we define two systems of affine equations associated to a t-motive $M$, used for calculation of $H^1(M)$ and $H_1(M)$. We describe the process of elimination of unknowns in these systems. This is an analog of the corresponding theory of systems of linear differential equations. It gives us a notion of a non-commutative determinant $det_{i,c}(M)$ which belongs to the Anderson ring $\Bbb C_\infty[T,\tau]$ of non-commutative polynomials. Finally, we calculate $det_{i,c}(M)$ for $M=$ a Drinfeld module or its 1-dual. Also, some explicit calculations are made for Anderson t-motives of dimension $n$, rank $2n$. Some problems of future research are formulated.

Equations

$$\text{goth}P\cdot \text{goth}X=0$$

$$\tau (\sum _{i=0}^{\infty }{a}_i{T}^i):=(\sum _{i=0}^{\infty }{a}_i{T}^i{)}^{(1)}:=\sum _{i=0}^{\infty }{a}_i^q{T}^i$$

$$P=\sum _{\gamma =0}^{r_0}{a}_{\gamma }{\tau }^{\gamma }+\sum _{\beta =1}^n\sum _{\gamma =0}^{{\varkappa }_{\beta }}{b}_{\beta \gamma }{\tau }^{\gamma }{T}^{\beta }\in {\mathbb{C}}_{\infty }[T,\tau ]$$

$$\sum _{j=1}^{\mu }{P}_{ij}{X}_j=0$$

$$D_{ij}:=\sum _{\alpha }{a}_{ij\alpha }{\left(\frac{d}{dx}\right)}^{\alpha }$$

$$\sum _{j=1}^{\mu }{D}_{ij}(X_j)=0.$$

$$P_{ij}=\sum _{\beta ,\gamma \ge 0}{a}_{ij;\beta \gamma }{\tau }^{\gamma }{T}^{\beta }$$

$$R_i:=\sum _{j=1}^{\mu }\sum _{\gamma =0}^{r_{ij0}}{a}_{ij;0\gamma }{x}_{j0}^{q^{\gamma }}=0$$

$${\tilde{R}}_1=0,\dots ,{\tilde{R}}_{\delta }=0,R_{\delta +1}=0,\dots ,R_{\mu }=0$$

$${\tilde{m}}_1:=m_1-1,\dots ,{\tilde{m}}_{\delta }:=m_{\delta }-1,m_{\delta +1},\dots ,m_{\mu }$$

$$\forall j=2,\dots ,nwe\; have\sum _{i=1}^n{C}_i{P}_{ij}=0.$$

$$C_i=\sum _{j=0}^{k(n-1)}{z}_{ij}{\tau }^j$$

$$(kn+1)(n-1)\times n(k(n-1)+1)$$

$$de{t}_{1,c}(\text{goth}P)X=0$$

$$P_{21}X=P_{22}X=0⟹X=0.$$

$$PX=K$$

$$\begin{array}{c}{P}_{12}{X}_2=-P_{11}{X}_1\\ P_{22}{X}_2=-P_{21}{X}_1\end{array}$$

$$\begin{array}{c}{P}_{12}{X}_2+...+P_{1n}{X}_n=-P_{11}{X}_1\\ \dots \\ P_{n2}{X}_2+...+P_{nn}{X}_n=-P_{n1}{X}_1\end{array}$$

$$de{t}_{2,c}(\widehat{\text{goth}}{P}_{i1})X_2=K$$

$$[(\text{Cal}M{)}_{\beta \gamma }{]}_{\delta \nu }=a_{\gamma \beta ,\delta -\nu }^{(\nu -1)}if\delta -\nu \in [0,\dots ,k]and[(\text{Cal}M{)}_{\beta \gamma }{]}_{\delta \nu }=0\; otherwise,$$

$$\left(\begin{array}{ccccccccc}{a}_{\gamma \beta 0}& 0& 0& \dots & 0& 0& \dots & 0& 0\\ a_{\gamma \beta 1}& a_{\gamma \beta 0}^{(1)}& 0& \dots & 0& 0& \dots & 0& 0\\ a_{\gamma \beta 2}& a_{\gamma \beta 1}^{(1)}& a_{\gamma \beta 0}^{(2)}& \dots & 0& 0& \dots & 0& 0\\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ a_{\gamma \beta k}& a_{\gamma \beta ,k-1}^{(1)}& a_{\gamma \beta ,k-2}^{(2)}& \dots & a_{\gamma \beta 0}^{(k)}& 0& \dots & 0& 0\\ & & & & & & & & \\ 0& a_{\gamma \beta k}^{(1)}& a_{\gamma \beta ,k-1}^{(2)}& \dots & a_{\gamma \beta 1}^{(k)}& a_{\gamma \beta 0}^{(k+1)}& \dots & 0& 0\\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ 0& 0& 0& \dots & 0& 0& \dots & a_{\gamma \beta 0}^{(k(n-1)-1)}& 0\\ & & & & & & & & \\ 0& 0& 0& \dots & 0& 0& \dots & a_{\gamma \beta 1}^{(k(n-1)-1)}& a_{\gamma \beta 0}^{(k(n-1))}\\ & & & & & & & & \\ 0& 0& 0& \dots & 0& 0& \dots & a_{\gamma \beta 2}^{(k(n-1)-1)}& a_{\gamma \beta 1}^{(k(n-1))}\\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ 0& 0& 0& \dots & 0& 0& \dots & a_{\gamma \beta k}^{(k(n-1)-1)}& a_{\gamma \beta ,k-1}^{(k(n-1))}\\ & & & & & & & & \\ 0& 0& 0& \dots & 0& 0& \dots & 0& a_{\gamma \beta k}^{(k(n-1))}\end{array}\right)$$

$$R_p(P_1,P_2):=\left|\begin{array}{cccccc}{a}_n& \dots & a_0& 0& \dots & 0\\ \dots & \dots & \dots & \dots & \dots & \dots \\ 0& \dots & 0& a_n^{q^{m-1}}& \dots & a_0^{q^{m-1}}\\ b_m& \dots & b_0& 0& \dots & 0\\ \dots & \dots & \dots & \dots & \dots & \dots \\ 0& \dots & 0& b_m^{q^{n-1}}& \dots & b_0^{q^{n-1}}\end{array}\right|$$

$$\text{goth}P=\left(\begin{array}{cc}{A}_0+A_1\tau & B_0+B_1\tau \\ C_0+C_1\tau & D_0+D_1\tau \end{array}\right)$$

$$de{t}_{1,c}(\text{goth}P)=\left|\begin{array}{cc}{A}_0& B_0\\ C_0& D_0\end{array}\right|\cdot {\left|\begin{array}{cc}{B}_0& B_1\\ D_0& D_1\end{array}\right|}^{(1)}+$$

$$+\left[\left|\begin{array}{cc}{A}_1& B_0\\ C_1& D_0\end{array}\right|\cdot {\left|\begin{array}{cc}{B}_0& B_1\\ D_0& D_1\end{array}\right|}^{(1)}+{\left|\begin{array}{cc}{A}_0& B_1\\ C_0& D_1\end{array}\right|}^{(1)}\cdot \left|\begin{array}{cc}{B}_0& B_1\\ D_0& D_1\end{array}\right|\right]\tau +{\left|\begin{array}{cc}{A}_1& B_1\\ C_1& D_1\end{array}\right|}^{(1)}\cdot \left|\begin{array}{cc}{B}_0& B_1\\ D_0& D_1\end{array}\right|{\tau }^2$$

$$\begin{array}{c}{a}_0{X}_1+a_1{X}_1^{\prime }+b_0{X}_2+b_1{X}_2^{\prime }=0\\ \\ c_0{X}_1+c_1{X}_1^{\prime }+d_0{X}_2+d_1{X}_2^{\prime }=0\end{array}$$

$$\text{Cal}M=\left(\begin{array}{cccc}{b}_0& b_0^{\prime }& d_0& d_0^{\prime }\\ b_1& b_0+b_1^{\prime }& d_1& d_0+d_1^{\prime }\\ 0& b_1& 0& d_1\end{array}\right)$$

$$(g_0+g_1\frac{d}{dx})(a_0+a_1\frac{d}{dx})+(h_0+h_1\frac{d}{dx})(c_0+c_1\frac{d}{dx})$$

$$de{t}_{1,c}(A)=C_1\cdot \text{goth}D(A)$$

$$M[[T]{]}^{\tau }=\{Y\in {\mathbb{C}}_{\infty }[[T]{]}^r|Y^{(1)}Q=Y\}$$

Full text

††support: Thanks: The authors are grateful to V. Retakh who indicated them the contents of Section 1.10 — relations of non-commutative determinants of the present paper and earlier notions of quasideterminants.

0. Introduction. The Anderson ring $\mathbb{C}_{\infty}[T,\tau]$ (see below) is an analog of the ring of linear differential operators, hence the below theory has a differential operators case analog (referred below as a ”D-case”). Surprisingly, apparently the analog of the approach of the present paper to the D-case was not published earlier; at least, it is not broadly known. We give in Remark 1.9.5 the first construction of this analog for the D-case. 0.1. Let $M$ be an Anderson t-motive. We consider a system of affine equations defining $H^{1}(M)$, and (another) system of affine equations defining $H_{1}(M)$. The system defining $H_{1}(M)$ is the system defining $T_{\goth}T(M)$ — the $\goth T$-Tate module of $M$, where $\goth T$ is the prime ideal of $\mathbb{F}_{q}[T]$ generated by $T$. See [NP] for a similar result. Further, the system defining $H^{1}(M)$ is the system defining $T_{\goth}T(M^{\vee})$, where $M^{\vee}$ is the 0-dual of $M$, see [HJ] for its definition. There is a strange phenomenon: apparently, the $T$-divisible modules of solutions to the systems of affine equations defining first, $H_{1}(M)$, and second, $H^{1}(M)$, are distinct as $\hbox{Gal }(\overline{\mathbb{F}_{q}(\theta)}/\mathbb{F}_{q}(\theta)$-modules (see (3.5): an example for the case of Drinfeld modules of rank 2), although if $M$ is uniformizable then $H_{1}(M)$ and $H^{1}(M)$ are in canonical perfect duality. To understand a reason of this distinction is a subject of future research. Most likely we should find an equivalence relation on the set of systems of affine equations, as well as of the sets of these solutions, making them dual. 0.1.1. Since the dimension of the set $T_{\goth}T(M)_{s}$ — the set of small elements (= sequences that tend to 0, see the definition below (2.2)) of $T_{\goth}T(M)$, is $h_{1}(M)$, it is natural to ask what is the dimension of the set of small elements of $T_{\goth}p(M)$ for all prime ideals $\goth p$ of $\mathbb{F}_{q}[T]$. The answer is simple: it is the same $h_{1}(M)$ for all $\goth p$. Finally, there exists a natural embedding of $H_{1}(M)$ to $Lie(M)=\mathbb{C}_{\infty}^{n}$ where $n$ is the dimension of $M$. It defines a Hodge-Pink structure on $M$. Since $H_{1}(M)=T_{\goth}T(M)_{s}$, we have an embedding $T_{\goth}T(M)_{s}\hookrightarrow Lie(M)$. 0.1.2. Do exist analogous embeddings of $T_{\goth}p(M)_{s}$ for other prime ideals $\goth p$? Generally, the present paper contains more questions than answers. 0.2. We use the standard definitions and notations for Anderson t-motives, see, for example, [A86], [G96], [GL20]. Namely, let $q$ be a power of a prime $p$, $\theta$ a transcendental element, $\mathbb{R}_{\infty}:=\mathbb{F}_{q}((\theta^{-1}))$, $\mathbb{C}_{\infty}:=\widehat{\overline{\mathbb{R}_{\infty}}}$ an algebaically closed complete field with a valuation $v$ given by $v(\theta)=-1$. The Anderson ring $\mathbb{C}_{\infty}[T,\tau]:=\mathbb{C}_{\infty}[T]\{\tau\}$ is a ring of non-commutative polynomials in two variables $T$ and $\tau$, and an Anderson t-motive $M$ is a left module over $\mathbb{C}_{\infty}[T,\tau]$ satisfying some properties. We shall consider systems of equations

[TABLE]

(see (1.2)) where $\goth P\in M_{\lambda\times\mu}(\mathbb{C}_{\infty}[T,\tau])$ and $\goth X\in M_{\mu\times 1}(\mathbb{C}_{\infty}[[T]])$ a column of power series. We shall develop a theory of these systems generalizing the corresponding theory of [O33.1], [O33.2] and of [G96], Section 1 for the case $\lambda=\mu=1$ and entries of $\goth P$, resp. of $\goth X$ belong to $\mathbb{C}_{\infty}\{\tau\}$, resp. $\mathbb{C}_{\infty}$. Particularly, we describe an elimination of variables reducing the system (0.2.1) to the case $\lambda=\mu=1$, and we find the rank of the set of solutions for $\lambda=\mu=n$. Some questions remain unsolved. What is an exact condition that the projection of $\goth W$ — the set of solutions to (0.2.1) — to $\goth W_{1}$ — the set of solutions of a reduced system — is an isomorphism? What is a condition that two systems of type (0.2.1) have a non-zero intersection of the sets of their solutions (to treat both cases $\lambda=\mu=1$ and $\lambda,\ \mu$ arbitrary)? Recall that for the case treated in [G86], [O33] this is a condition that the $p$-resultant is $\neq 0$. See [E08] for a formula of $p$-resultant similar to the formula for ordinary resultant. 1. Systems of affine equations. The ring $\mathbb{C}_{\infty}[[T]]$ is a left module over $\mathbb{C}_{\infty}[T,\tau]$, where the multiplication by $\tau$ is defined by the formula

[TABLE]

An affine equation is an equation $PX=0$ where $X=\sum_{i=0}^{\infty}x_{i}T^{i}\in\mathbb{C}_{\infty}[[T]]$ is an unknown and

[TABLE]

is a coefficient, see [GL21], (2.2.2). See also [GL21], (2.1) - (2.1.2) for a form of an affine equation as a system of ordinary polynomial equations defining consecutively $x_{0},\ x_{1},\dots$, where $x_{i}\in\mathbb{C}_{\infty}$. Definition 1.1.1. A sequence $x_{0},\ x_{1},\dots$, where $x_{i}\in\mathbb{C}_{\infty}$, is called a holonomic sequence, if $X:=\sum_{i}x_{i}T^{i}$ is a solution to an above equation $PX=0$ ($P\neq 0$). The terms corresponding to the $T$-free term $\sum_{\gamma=0}^{r_{0}}a_{\gamma}\tau^{\gamma}$ of (1.1) are called the head terms, other terms are called the tail terms. 1.1.2. $r_{0}$ is called the rank of $x_{0},\ x_{1},\dots$ and $n$ the length of its tail. We have $r_{0}$ is the degree of $P$ in $\tau$ (we assume that all $\varkappa_{i}<r_{0}$) and $n$ is the degree of $P$ in $T$. A system of $\lambda$ affine equations in $\mu$ variables $X_{1},\dots,X_{\mu}\in\mathbb{C}_{\infty}[[T]]$ has the $i$-th equation ($i=1,\dots,\lambda$) of the form (see (0.2.1))

[TABLE]

where coefficients $P_{ij}\in\mathbb{C}_{\infty}[T,\tau]$ are as above. They form a $\lambda\times\mu$-matrix denoted by $\goth P$. Remark 1.2.1. We can consider a version of (1.2) for the case $P_{ij}\in\mathbb{C}_{\infty}[[T]]\{\tau\}$, as well as for $P_{ij}\in\mathbb{C}_{\infty}[[T]]\{\{\tau\}\}$. Remark 1.2.2. For the D-case we have: let $a_{ij\alpha}$ be functions,

[TABLE]

differential operators, $X_{j}$ unknown functions. The $i$-th equation of an analog of (1.2) is

[TABLE]

The set of solutions to (1.2) is a $\mathbb{F}_{q}[[T]]$-submodule of $(\mathbb{C}_{\infty}[[T]])^{\mu}$. We denote it by $\goth W=\goth W(\goth P)$. It is $T$-divisible, i.e. if $X=(X_{1},\dots,X_{\mu})\in\goth W$ and $X/T\in(\mathbb{C}_{\infty}[[T]])^{\mu}$ then $X/T\in\goth W$. See [GL21], (2.2.3) for the case $\lambda=\mu=1$. Let $\lambda=\mu$. The $\mathbb{F}_{q}[[T]]$-rank of $\goth W$ for the simplest case is the following. We let

[TABLE]

We denote by $r_{ij0}$ the maximal value of $\gamma$ such that $a_{ij;0\gamma}\neq 0$. Further, we denote $X_{i}=\sum_{\beta=0}^{\infty}x_{i\beta}T^{\beta}$. The $i$-th equation of (1.2) for the $T$-free terms is

[TABLE]

1.2.3. It is an additive polynomial in $x_{10},\dots,x_{\mu 0}$ of degree $q^{m_{i}}$ where $m_{i}:=\underset j\to{\hbox{max }}r_{ij0}$. Hence, if the intersections of the hypersurfaces $R_{i}=0$ are transversal at all points then the quantity of these intersection points is $q^{\sum_{i}m_{i}}$, and the $\mathbb{F}_{q}[[T]]$-rank of $\goth W$ is $\sum_{i=1}^{\mu}m_{i}$. 1.2.4. We denote $\Cal{A}:=(a_{ij;00})$. Since $\frac{\partial R_{i}}{\partial x_{j0}}=a_{ij;00}$ we get that if $|\Cal{A}|\neq 0$ then the rank of $\goth W$ is $\sum_{i}m_{i}$. Let $|\Cal{A}|=0$, i.e. the corank of $\Cal{A}$ is $\delta>0$. We consider linear transformations of lines of $\Cal{A}$ making its first $\delta$ lines the zero lines. We make the linear transformations of equations $R_{i}=0$ with the same coefficients. As a result, we get that the first $\delta$ of equations $R_{i}=0$ ($i=1,\dots,\delta$) will be without free terms, hence there exist $\tilde{R}_{i}\in\mathbb{C}_{\infty}\{\tau\}$ such that $R_{i}=\tilde{R}_{i}^{q}$. The system

[TABLE]

has the same set of solutions $\goth W$, and its $m_{i}$ will be

[TABLE]

and the problem of finding the rank of $\goth W$ will be reduced to the same problem for the system (1.3). If necessary we can repeat the same constriction several times. Hence, the case of general $\goth P$ can always be reduced to the case of $\goth P$ having $|\Cal{A}|\neq 0$. Question 1.4. Let $\goth W\subset(\mathbb{C}_{\infty}[[T]])^{\mu}$ be a $T$-divisible $\mathbb{F}_{q}[[T]]$-submodule. How to restitute $\goth P$ by $\goth W$? For the case $\lambda=\mu=1$ we can restitute $P$ uniquely (under some conditions), see [GL21], Proposition 2.3.2.

Let now $\lambda=\mu=n$. For a generic $\goth P$ we can eliminate $X_{2},\dots,X_{n}$ using cofactors, like in the commutative case. Proposition 1.5. For a generic quadratic $\goth P$ there exist non-zero cofactors $C_{1},\dots,C_{n}\in\mathbb{C}_{\infty}[T,\tau]$ such that

[TABLE]

Proof. It is analogous to the proof for the D-case, see 1.9.5. Let $k$ be the maximal degree of all $P_{ij}$ as polynomials in $\tau$. We let

[TABLE]

where $z_{ij}\in\mathbb{C}_{\infty}[T]$ are indeterminate coefficients. Products $C_{i}P_{ij}$ are of degree $kn$ in $\tau$, hence for a fixed $j$ the equation $\sum_{i=1}^{n}C_{i}P_{ij}=0$ is a system of $kn+1$ linear equations in $z_{ij}$ with coefficients in $\mathbb{C}_{\infty}[T]$. Since $j=2,\dots,n$, we have $(kn+1)(n-1)$ linear equations. There are $n(k(n-1)+1)$ unknowns $z_{ij}$ - one more than equations.

Hence, we get a matrix denoted by $\Cal{M}=\Cal{M}(\goth P)$ of size

[TABLE]

with entries in $\mathbb{C}_{\infty}[T]$, of coefficients of this system of linear equations. The system is $\Cal{M}\cdot(z_{10},\dots,z_{n,k(n-1)})^{t}=0$ ($t$ means transposition). By definition, $z_{ij}$ are cofactors of $\Cal{M}$ and $C_{i}$ are from (1.5.1). $\square$ Definition 1.6. $\sum_{i=1}^{n}C_{i}P_{i1}$ is called the determinant of $\goth P$ along the first column. Notation: $det_{1,c}(\goth P)$. The same determinant for the $i$-th column is denoted by $det_{i,c}(\goth P)$. **Remark. ** If we choose $k>$ than the maximal degree of $P_{ij}$ as polynomials in $\tau$, then the last line of $\Cal{M}$ and hence all $C_{i}$ would be 0. 1.7. Clearly Proposition 1.5 and Definition 1.6 are valid for any commutative ring $R$ endowed with an automorphism $x\mapsto x^{(1)}$, and $R\{\tau\}$ satisfying $\tau x=x^{(1)}\tau$ for any $x\in R$. Namely, in notations of (1.2), $P_{ij}\in R\{\tau\}$, $X_{i}\in R$. In our case $R$ is $\mathbb{C}_{\infty}[T]$, and $X_{i}$ from (1.2) are from $\mathbb{C}_{\infty}[[T]]$. We denote the projection of $\mathbb{C}_{\infty}[[T]]^{n}\to\mathbb{C}_{\infty}[[T]]$ to its first coordinate by $\alpha=\alpha_{1}$, and the set of $X\in\mathbb{C}_{\infty}[[T]]$ satisfying

[TABLE]

by $\goth W_{1}=\goth W_{1}(\goth P)$. Obviously $\alpha(\goth W)\subset\goth W_{1}$. 1.8. Under some reasonable conditions the map $\alpha|_{\goth}W:\goth W\to\goth W_{1}$ is an isomorphism. Really, let $n=2$ and $P_{21}$, $P_{22}$ are ”coprime from the right”, i.e. for $X\in\mathbb{C}_{\infty}[[T]]$

[TABLE]

Then obviously $\alpha|_{\goth}W$ is a monomorphism. Let us consider epimorphisity. First, a $T$-divisible $\mathbb{F}_{q}[[T]]$-submodule of $\mathbb{C}_{\infty}[[T]]$ of finite rank has no proper $T$-divisible submodules of the same rank, because its elementary divisors are powers of $T$, which contradicts to the condition of $T$-divisibility. Further, let all $m_{i}$ from (1.2.3) be equal to $k$ from 1.5, and for $\Cal{A}$ from (1.2.4) we have $|\Cal{A}|\neq 0$. In this case the rank of $\goth W$ is $kn$. Proof of 1.5 shows that for a generic case the rank of $\goth W_{1}$ is also $kn$. It remains to prove that $\alpha(\goth W)$ is $T$-divisible. We restrict ourselves by the case $n=2$. Let $(XT,Y)\in\goth W$. We have $P_{12}\cdot Y,\ P_{22}\cdot Y\in\mathbb{C}_{\infty}[[T]]\cdot T$. For a generic case, $\sum_{\gamma}a_{12;0\gamma}\tau^{\gamma}$ and $\sum_{\gamma}a_{22;0\gamma}\tau^{\gamma}$ are coprime from the right. This implies that $Y\in\mathbb{C}_{\infty}[[T]]\cdot T$, hence $(X,Y/T)\in\goth W$ and $X\in\alpha(\goth W)$. Explicitly, we can describe $\alpha^{-1}$ as follows. Let us consider an equation

[TABLE]

where $P\in\mathbb{C}_{\infty}[T,\tau]$ is a coefficient, $X\in\mathbb{C}_{\infty}[[T]]$ an unknown and $K\in\mathbb{C}_{\infty}[[T]]$ a free term. Let $P=\sum_{i}P_{i}T^{i}$ where $P_{i}\in\mathbb{C}_{\infty}\{\tau\}$. If $P_{0}\neq 0$ then (1.8.1) has solutions. Let $n=2$ and $X_{1}\in\goth W_{1}(\goth P)$. To find $X_{2}$ such that $(X_{1},\ X_{2})\in\goth W$, we should solve a system

[TABLE]

Both these equations are of type (1.8.1). Since $X_{1}\in\goth W_{1}$ and $\alpha|_{\goth}W$ is an epimorphism, we get that they have a common root. If $n>2$ then the system (1.8.2) has the form

[TABLE]

Let us fix $i\in[1,\dots,n]$. We denote by $\hat{\goth}P_{i1}$ the submatrix of $\goth P$ obtained by elimination of the $i$-th line and first column. We disregard the $i$-th equation of (1.8.3) and find $X_{2}$ satisfying the remaining equations using the method of elimination of $X_{3},\dots,X_{n}$ of Proposition1.5. Namely, we get an equation

[TABLE]

where $K\in\mathbb{C}_{\infty}[[T]]$ is a linear combination of $P_{j1}X_{1}$ ($j\neq i$) with cofactors. Since $X_{1}\in\goth W_{1}$ and $\alpha|_{\goth}W$ (we hope) is an epimorphism, we get that all (1.8.4) have a common root. Continuing the process we get $(X_{1},X_{2},\dots,X_{n})\in\goth W$ over $X_{1}$. In some particular cases (see Section 5, $n=2$: for this case $P_{12}$ is invertible in a larger ring) it is easy to get a proof that in these cases $\alpha|_{\goth}W$ is an isomorphism. A proof for a general case is a problem of further research. 1.9. Let us give the explicit form of $\Cal{M}$. We denote $P_{ij}=\sum_{\alpha=0}^{k}a_{ij\alpha}\tau^{\alpha}$ where $a_{ij\alpha}\in\mathbb{C}_{\infty}[T]$ are coefficients. $\Cal{M}$ is a block matrix: $\Cal{M}=(\Cal{M})_{\beta\gamma}$ where $\beta=2,\dots,n$, $\gamma=1,\dots,n$, and all $(\Cal{M})_{\beta\gamma}$ are of size $(kn+1)\times(k(n-1)+1)$. The $(\delta,\nu)$-th entry of $(\Cal{M})_{\beta\gamma}$ is

[TABLE]

where for $A=\sum_{i=0}^{\infty}y_{i}T^{i}\in\mathbb{C}_{\infty}[T]$ we denote $A^{(k)}:=\sum_{i=0}^{\infty}y_{i}^{q^{k}}T^{i}$. For example, we have (case $n\geq 4$): $(\Cal{M})_{\beta\gamma}=$

[TABLE]

An analog of (1.9.1) for the D-case has slighly other form, see (1.9.7) below. Remark 1.9.2. Recall the definition of the $p$-resultant (see, for example, [E08]). Let $P_{1}=\sum_{i=0}^{n}a_{i}\tau^{i},\ P_{2}=\sum_{i=0}^{m}b_{i}\tau^{i}\in\mathbb{C}_{\infty}\{\tau\}$. Then (rough form; really, the structure of this matrix is the same as the structure of ordinary resultant: there are $m$ lines of $a_{*}$’s and $n$ lines of $b_{*}$’s):

[TABLE]

We see that the above blocks (1.9.1) are blocks of $R_{p}$. Example 1.9.3: $n=2,\ k=1$,

[TABLE]

The corresponding determinant is

[TABLE]

[TABLE]

For the case $n=2$, $k>1$, $\goth P=\left(\begin{matrix}A&B\\ C&D\end{matrix}\right)\in M_{2\times 2}\mathbb{C}_{\infty}\{\tau\}$ the $\tau$-free term of $det_{1,c}(\goth P)$ is $\left|\begin{matrix}A_{0}&B_{0}\\ C_{0}&D_{0}\end{matrix}\right|\cdot R_{p}(B,D)^{(1)}$. Remark 1.9.5. Recall that the analog of 1.9.3 (i.e. $n=2,\ k=1$) for the D-case is the following. Let $a_{*},\ b_{*},\ c_{*},\ d_{*}$ be functions which are coefficients, and $X_{1},X_{2}$ unknowns. Analog of (1.2) is

[TABLE]

In order to eliminate $f_{2}$, we apply a differential operator $g_{0}+g_{1}\frac{d}{dx}$, resp. $h_{0}+h_{1}\frac{d}{dx}$ to the first, resp. second equation of (1.9.6), where $g_{0},\ g_{1},\ h_{0},\ h_{1}$ are indeterminates. We want that after summation $f_{2}$ disappears. We have

[TABLE]

and $g_{*},\ h_{*}$ satisfy $\Cal{M}\cdot(g_{0},\ g_{1},\ h_{0},\ h_{1})^{t}=0$, hence $g_{*},\ h_{*}$ are cofactors of $\Cal{M}$. The second order operator

[TABLE]

annulating $X_{1}$, is the 1-determinant of the system. **1.10. Relations with other quasideterminants. ** Let $A=\left(\begin{matrix}a_{11}&a_{12}\\ a_{21}&a_{22}\end{matrix}\right)$ be any matrix with entries in a non-commutative ring. Its quasideterminant $\goth D(A)$ is $a_{11}-a_{12}a_{22}^{-1}a_{21}$ (see, for example, [GGRW05]). Let $C_{1}$, $C_{2}$ be from 1.5, i.e. $C_{1}a_{12}+C_{2}a_{22}=0$. This implies $a_{12}a_{22}^{-1}=-C_{1}^{-1}C_{2}$ and $\goth D(A)=a_{11}+C_{1}^{-1}C_{2}a_{21}$ hence

[TABLE]

Similar formulas exist for $A$ of size $>2$. 2. Systems of affine equations associated to Anderson t-motives. Let $M$ be an Anderson t-motive of rank $r$. We choose a basis $f_{*}:=(f_{1},\dots,f_{r})^{t}$ of $M$ over $\mathbb{C}_{\infty}[T]$. We denote by $Q\in M_{r\times r}(\mathbb{C}_{\infty}[T])$ the matrix of the action of $\tau$ on $f_{*}$, i.e. $\tau f_{*}=Qf_{*}$. We denote M[[T]]:=M\underset\mathbb{C}_{\infty}[T]\to{\otimes}\mathbb{C}_{\infty}[[T]]. $\tau$ acts on $M[[T]]$. We identify $M[[T]]$ with $\mathbb{C}_{\infty}[[T]]^{r}$ using the basis $f_{*}$, and under this identification (here $Y$ is a row vector)

[TABLE]

(2.1) is a system of affine equations

[TABLE]

According [A86], we have: $H^{1}(M)$ is the set of small solutions to (2.2) (small solutions are solutions whose coefficients tend to 0 — see [GL21], above Proposition 2.3.3. The small rank is the rank of the module of small solutions). In order to solve (2.2), we use the above elimination process. Let $P=P_{1}(M):=det_{1,c}(Q^{t}\tau-I_{r})$. We use notations of (1.1) for $P$. Conjecturally $r_{0}$ of (1.1) is $r$ — the rank of $M$. Also conjecturally, the small rank of solutions to (2.2) is equal to the small rank of solutions to $PX=0$. Recall that $n$ from (1.1) is called the length of the tail of $P$. We conjectured in [GL21] that it is equal to the dimension of $M$. This is true for standard t-motives having $n=2$, $r=4$ or 5. We shall show below that this is true for Drinfeld modules and their 1-duals. But for elementary t-motives of dimension $\geq 3$ this is wrong, see Section 5 below. 2.3. $H_{1}(M)$. All written above can be applied to $H_{1}(M)$ as well. Let

[TABLE]

We have $H_{1}(M)=\hbox{Hom}_{\mathbb{C}_{\infty}[T]}(M,\mathbb{C}_{\infty}\{T\})^{\tau}$ which is the set of small elements in $\hbox{Hom}_{\mathbb{C}_{\infty}[T]}(M,\mathbb{C}_{\infty}[[T]])^{\tau}$. We can identify column vectors $X\in\mathbb{C}_{\infty}[[T]]^{r}$ with elements of $\hbox{Hom}_{\mathbb{C}_{\infty}[T]}(M,\mathbb{C}_{\infty}[[T]])$. Namely, let $\varphi\in\hbox{Hom}_{\mathbb{C}_{\infty}[T]}(M,\mathbb{C}_{\infty}[[T]])$. The column vector $X$ corresponding to $\varphi$ is, by definition, $\left(\begin{matrix}\varphi(f_{1})\\ \dots\\ \varphi(f_{r})\end{matrix}\right)$. The analog of (2.1) is

[TABLE]

(2.3.1) is a system of affine equations (analog of (2.2) ):

[TABLE]

Let us recall a definition of $T_{\goth}T(M)$ — the $\goth T$-Tate module of $M$. Let $E(M)$ be the t-module corresponding to $M$. If we fix a basis $e_{1},\dots,e_{n}$ of $M$ over $\mathbb{C}_{\infty}\{\tau\}$ then we get an identification of $E(M)$ and $\mathbb{C}_{\infty}^{n}$. By definition, we have

[TABLE]

Proposition 2.3.4. The system (2.3.2) essentially coincides with the system (2.3.3) defining the Tate module $T_{\goth}T(M)$ (via equations $T(x_{i+1})=x_{i}$ ). Remark 2.3.5. To get an explicit form of equations $T(x_{i+1})=x_{i}$ defining $T_{\goth}T(M)$ we use a matrix of multiplication by $T$ in $e_{1},\dots,e_{n}$, while the above (2.3.2) uses a basis $f_{1},\dots,f_{r}$ of $M$ over $\mathbb{C}_{\infty}[T]$. The word ”essentially” in the statement of 2.3.4 means that the systems 2.3.2, 2.3.3 conicide after some elementary transformations. We can prove Proposition 2.3.4 for $M$ of any explicit form. We do not know whether there exists a proof which is independent on an explicit description of $M$, or not. 2.4. Duality and multiplication by the Carlitz module. Let $M$ be an Anderson t-motive. Its 0-dual $M^{\vee}$ is defined for example in [HJ], it is a more general object that a t-motive of [A86]: t-motives of [A86] are effective t-motives of [HJ]. By definition,

[TABLE]

We see that {(2.2) for $M$} is equal to {(2.3.2) for $M^{\vee}$}. This is in concordance with the canonical isomorphism $H^{1}(M)=H_{1}(M^{\vee})$. Let $\goth C$ be the Carlitz module. We have $Q(M\otimes\goth C^{\otimes n})=(T-\theta)^{n}Q(M)$. Let $\Xi=\sum_{i=0}^{\infty}\xi_{i}T^{i}$ be a solution to

[TABLE]

satisfying $\forall\ i\ \ v(\xi_{i+1})>v(\xi_{i})$ (see [G96], Example 5.9.36, p. 172). It is unique up to a factor from $\mathbb{F}_{q}^{*}$. We have $\Xi,\ \Xi^{-1}\in\mathbb{C}_{\infty}\{T\}$. Formula (2.1) implies that we have an isomorphism $H^{1}(M)\to H^{1}(M\otimes\goth C^{\otimes n})$ defined by $Y\mapsto\Xi^{n}Y$. It is canonical, i.e. it does not depend on a choice of $f_{1},\dots,f_{r}$. The same is true for $H_{1}(M)\to H_{1}(M\otimes\goth C^{\otimes n})$. Nevertheless, systems of affine equations (2.2) for $M$ and $M\otimes\goth C^{\otimes n}$ are quite different. For example, let $M^{\prime}$ be the 1-dual of $M$ (see [Tg95]; [GL07]). By definition, $M^{\prime}=M^{\vee}\otimes\goth C$; its advantage is that $M^{\prime}$ is an effective t-motive if $M$ is pure and has the nilpotent operator $N=0$. Let us consider the case of a Drinfeld module $M$. We compare the affine equations for $H^{1}(M^{\prime})$ (see (4.2), (4.4)) and the affine equation for $H_{1}(M)$, which coincide with the affine equation for $H^{1}(M^{\vee})$. We see that they are essentially different. Compare also (5.1) and (5.3) — case of $M$ of rank 4, dimension 2. Hence the operation of tensoring by $\goth C$, although it leaves $H^{1}$ invariant, modifies the affine equation. We can ask what will occur with affine equations after other tensor products and Hom’s, not only after tensoring by $\goth C$. 2.5. Problems of further research. A. For any types of t-motives find the set of bidegrees $\Cal{D}$, i.e. the set of pairs $(\beta,\gamma)\in(\mathbb{Z}^{+})^{2}$ such that $b_{\beta\gamma}\neq 0$ (see (1.1)), as functions of entries of $Q$. B. Is it possible to define duality on the set of systems of affine equations (or on its quotient set, by some appropriate equivalence relation) coming from the duality of Anderson t-motives, or, the same, duality between $H^{1}(M),\ H_{1}(M)$? See (3.5.2), (3.5.3) as an example of such duality for Drinfeld modules of rank 2. C. We can find more analogs of the theorems of the D-case. For example, the member-by-member product of two holonomic sequences is also a holonomic sequence (D-case). Is the analogous property true for the case of t-motives? Namely, let $x_{0},x_{1},\dots$ and $y_{0},y_{1},\dots$ be solutions to equations $P_{1}X=0$, $P_{2}X=0$ (see Remark 1.1.1), where $P_{1}$, $P_{2}$ are of finite tail. Is the sequence $x_{0}y_{0},x_{1}y_{1},\dots$ a solution of such equation? D. Galois action. Let $K$ be a finite extension of $\mathbb{F}_{q}(\theta)$. If coefficients of $P_{ij}$ from (1.2) are in $K$ then the Galois group $\hbox{Gal }(K):=\hbox{Gal }(\bar{K}/K)$ acts on $\goth W$ — the set of solutions to (1.2). Hence, if $\lambda=\mu=r$ we have a map

[TABLE]

Remark 2.5.D.2. $\hbox{Gal }(K)$ does not preserve small elements. Really, there are many inclusions $i:\bar{K}\hookrightarrow\mathbb{C}_{\infty}$. The restriction of the valuation $v$ on $\mathbb{C}_{\infty}$ with respect to an inclusion induces a valuation $i^{*}(v)$ on $\bar{K}$. Let us fix an inclusion $i_{0}:\bar{K}\hookrightarrow\mathbb{C}_{\infty}$. An element $\sum_{j=0}^{\infty}a_{j}T^{j}$, where $a_{j}\in\bar{K}$, is small, if $\underset j\to\infty\to{\lim}v(i_{0}(a_{j}))=+\infty$. Equivalently, $\underset j\to\infty\to{\lim}[i_{0}^{*}(v)](a_{j})=+\infty$. For $\gamma\in\hbox{Gal }(K)$ we have $[\gamma(i_{0}^{*}(v))](\gamma(v))=[i_{0}^{*}(v)](a_{j})$, hence $\underset j\to\infty\to{\lim}[\gamma(i_{0}^{*}(v))](\gamma(v))=+\infty$ but not necessarily $\underset j\to\infty\to{\lim}[i_{0}^{*}(v)](\gamma(v))=+\infty$, i.e. $\hbox{Gal }(K)$ does not preserve small elements. We can conjecture that the standard properties of $\goth p$-adic representations hold for $\rho$. Namely, for almost all $P_{ij}$ we have: im $\rho$ is of finite index in $GL_{r}(\mathbb{F}_{q}[[T]])$. Further, let $\goth p$ be a prime of $K$. We denote by $\bar{\goth}p$ an extension of $\goth p$ to $\bar{K}$ and by $G_{i}\subset G_{d}\subset Gal(K)$ the inertia and decomposition subgroups. Like for the case of the Galois action on Tate modules, we expect that for almost all $\goth p$ we have: $G_{i}$ acts trivially on $\goth W$, hence the action of $Fr_{\goth}p$ on $\goth W$ is defined up to a conjugation. What is known on its characteristic polynomial? E. (See also 0.1.1, 0.1.2). Let $\goth p\subset\mathbb{F}_{q}[T]$ be a prime ideal. Elements of the Tate module $T_{\goth}p(M)$ are sequences $x_{0},\ x_{1},\dots$, where $x_{i}\in E(M)=\mathbb{C}_{\infty}^{n}$, $x_{0}=0$, such that $\goth p(x_{i})=x_{i-1}$. They are defined by systems of affine equations. We can define $T_{\goth}p(M)_{s}$. What is their small rank? Does it depend on $\goth p$? As a first example, we can consider the case of $\goth p$ of degree 1, i.e. $\goth p=T+c$ for $c\in\mathbb{F}_{q}$. Further, the identification of $T_{\goth}p(M)_{s}$ and $H_{1}(M)$ shows that there is a natural inclusion of $T_{\goth}p(M)_{s}$ to $Lie(M)$. Does there exist something similar for $T_{\goth}p(M)_{s}$? 3. Case of Drinfeld modules. Let $M$ be a Drinfeld module of rank $r$. It is free of rank 1 over $\mathbb{C}_{\infty}\{\tau\}$. Let $e$ be the only element of a basis of $M$ over $\mathbb{C}_{\infty}\{\tau\}$ such that the action of $T$ on $e$ is given by the formula (here $a_{i}\in\mathbb{C}_{\infty}$ are parameters):

[TABLE]

We have: a column $f_{*}=(e,\ \tau e,\ \tau^{2}e,\dots,\tau^{r-1}e)^{t}$. Its $Q$ is

[TABLE]

and its $\goth P=Q^{t}\tau-I_{r}$ is

[TABLE]

We find $det_{r,c}(\goth P)$ ”manually”, i.e. by explicit elimination of unknowns. We denote (compare with [GL21], Section 3): $Y=(y_{0}\ y_{1}\ \dots\ y_{r-1})$ (caution: shift of indices), where $y_{i}\in\mathbb{C}_{\infty}[[T]]$. (2.1) becomes

[TABLE]

We denote (like in [GL21], the line below (3.8))

[TABLE]

where $x_{i}\in\mathbb{C}_{\infty}$, and we substitute this formula to the equations of (3.1):

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Comparing (3.2) and (3.3) we get an affine equation to find $x_{0},\ x_{1},\ x_{2},\dots$:

[TABLE]

[TABLE]

[TABLE]

and for any $i$

[TABLE]

hence

[TABLE]

(here $a_{0}=\theta$, $a_{r}=1$). We see that the length of the tail is 1; the tail consists of one term $-\ x_{i-1}^{q^{r}}$. The set of bidegrees $\Cal{D}$ from 2.5.A is $(0,0);\ (0,1);\dots;(0,r);\ (1,r)$. 3.5. The affine equation for $H_{1}(M)$ coincides with the affine equation for $T_{\goth}T(M)$ (Proposition 2.3.4). It is (we omit details of calculation):

[TABLE]

Apparently it differs essentially from the above equation (3.4.1). For example, for $r=2$, $i=0$ the equations (3.5.1), resp. (3.4.1) are

[TABLE]

and

[TABLE]

Equations (3.5.3) and (3.5.2) are different: there exist $a_{1}\in\mathbb{F}_{q}(\theta)$ such that (3.5.3) has a non-trivial root in $\mathbb{F}_{q}(\theta)$ while (3.5.2) has no such root, and vice versa. For example, there exist $a_{1}$ such that the Galois groups of (3.5.2), (3.5.3) are different. 4. Case of duals of Drinfeld modules. Let $M$ be as above. Its 1-dual $M^{\prime}$ has rank $r$ and dimension $r-1$. We have $Q(M^{\prime})=(T-\theta)[Q(M)^{-1}]^{t}$:

[TABLE]

and

[TABLE]

As above we make an elimination ”manually”. We denote (shift in 1 in indices): $Y=(y_{1}\ y_{2}\ \dots\ y_{r})$. (2.1) becomes

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

We denote $N_{k}:=(T-\theta^{q^{k}})(T-\theta^{q^{k-1}})\cdot...\cdot(T-\theta^{q^{2}})(T-\theta^{q})$, $N_{0}:=1$, and we denote its coefficients by $v_{ki}$ (they are polynomials in $\theta$):

[TABLE]

(4.1) implies $y_{k}^{(1)}=N_{k-1}\ y_{1}^{(k)}$. We denote like in (3.2): $y_{1}=x_{0}+x_{1}T+x_{2}T^{2}+...$. The first equation of (4.1) becomes the following constituents of an affine equation:

[TABLE]

(the head — the 0-th equality of the affine equation);

[TABLE]

[TABLE]

(the first equality of the affine equation);

[TABLE]

[TABLE]

[TABLE]

(the second equality of the affine equation); etc. The first equation of the maximal length of the tail is the $(r-1)$-th equality in the affine equation:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Finally, we write the $k$-th equality in the affine equation for $k\geq r-1$:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Here $a_{r}=1$, $i$ is the number of a line in (4.2), and $j$ the number of a term in the $i$-th line, both $i,\ j$ are counted from 0. Equivalently, we have

[TABLE]

The set of bidegrees $\goth D$ is the union of a triangle with vertices $(0,1);\ (0,r);\ (r-1,r)$ and a point (0,0). We see that the length of the tail is $r-1$. 4.3. Let us transform (4.2) to the form where $\varkappa_{\beta}<r$ (here $\varkappa_{\beta}$ are from (1.1)), in order to show that the length of the tail of the corresponding affine equation is $r-1$. We restrict ourselves by the case $r=3$. Formula (4.2) for $k=0$ is (we have $v_{00}=v_{11}=v_{22}=1$):

[TABLE]

[TABLE]

Formula (4.2) for $k=1$ is

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Formula (4.2) for $k=2$ is

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Substituting values of $v_{10},\ v_{20},\ v_{21}$ we get that the coefficients of the last line of (4.4) are non-0, hence the corresponding affine equation has the length of tail 2. 5. The elementary t-motives. Let us consider the following Anderson t-motives $M$ of dimension $n$, rank $r=2n$. Let $e_{*}=(e_{1},\dots,e_{n})^{t}$ be a basis of $M$ over $\mathbb{C}_{\infty}\{\tau\}$. The action of $T$ on $e_{*}$ is defined by the formula

[TABLE]

where $A\in M_{n\times n}(\mathbb{C}_{\infty})$ is a matrix parameter. This t-motive is denoted by $M(A)$. For $n=1$ it is a Drinfeld module of rank 2. We shall call them elementary t-motives. We choose a basis $f_{*}=(e_{1},\dots,e_{n},\tau e_{1},\dots,\tau e_{n})^{t}$ of $M$ over $\mathbb{C}_{\infty}[T]$. The matrix $Q$ in $f_{*}$ is $\left(\begin{matrix}0&I_{n}\\ (T-\theta)\ I_{n}&-A\end{matrix}\right)$ (block form, entries are $n\times n$-matrices). Hence,

[TABLE]

Making elementary transformations (formalization of this construction is a research problem) we reduce $\goth P_{2n}$ to the following matrix $\goth P_{n}=((T-\theta^{q})\tau^{2}-1)\cdot I_{n}-A^{t}\tau$, i.e.

[TABLE]

Formulas (1.5.1) for this case become

[TABLE]

The matrices $\Cal{M}$ are:

[TABLE]

[TABLE]

[TABLE]

(its block structure is denoted by spaces).

For $n=2$ we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

We see that this is [GL21], (3.9), up to a constant factor. (3.11) of [GL21] is the linear combination $C_{1}P_{11}+C_{2}P_{21}$. The first equation of $\goth P_{2}\left(\begin{matrix}X_{1}\\ X_{2}\end{matrix}\right)=0$ gives us

[TABLE]

This shows that for this case $\alpha|_{\goth}W:\goth W\to\goth W_{1}$ (see (1.8) ) is an isomorphism. Remark 5.2. Analog of (5.1) for $H_{1}(M)$ is

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Description of a symmetry between (5.1) and (5.3) is an exercise. This is an analog for $n=2$ of a symmetry between (3.5.2) and (3.5.3). First, we should transpose $A$, because $M(A)^{\prime}=M(A^{t})$. Research problems. We described in [GL07] a class of t-motives called standard-2 t-motives. They are t-motives for which bases of $M$ over both $\mathbb{C}_{\infty}[T],\ \mathbb{C}_{\infty}\{\tau\}$ have a simple description. We want to find the set of bidegrees $\Cal{D}$ (see 2.5A) for their $det_{1,c}(\goth P)$. Further, there is a conjecture that for almost all t-motives $M$ of rank $r$ we have $h^{1}(M)$ is either 0 or $r$. The above theory of elimination permits us to verify this conjecture by explicit calculations. 6. Properties of holonomic sequences. Analogs of closure properties of holonomic sequences in the D-case (see, for example, [K], Theorem 4; [KP], Theorem 7.2) hold in the present case. Let us recall some properties for the D-case. Let $D:\ \mathbb{C}[[T]]\to\mathbb{C}[[T]]$ be the operator of derivation. Definition 6.1. A sequence $x_{0},\ x_{1},\dots$, where $x_{i}\in\mathbb{C}$, is called a holonomic sequence, if two equivalent conditions hold: 6.1.2. $\exists\ r_{1}>0$ and polynomials $p_{0}(x),\dots,p_{r_{1}}(x)\neq 0$ such that $\forall\ k\geq 0$ we have

[TABLE]

6.1.3. $\exists\ r_{2}>0$ and polynomials $q_{0}(T),\dots,q_{r_{2}}(T)\neq 0$ such that $f(T):=\sum_{i=0}^{\infty}x_{i}T^{i}\in\mathbb{C}[[T]]$ satisfies $P(f)=0$, where $P=\sum_{i=0}^{r_{2}}q_{i}D^{i}$ is a differential operator, i.e.

[TABLE]

Let $n_{1}$, resp. $n_{2}$ be the maximal degree of $p_{i}(x)$, resp. $q_{i}(T)$. We have ([KP], Th. 7.1): If (6.1.2) holds for $x_{0},\ x_{1},\dots$ with some $r_{1}$, $n_{1}$, then (6.1.3) holds for $x_{0},\ x_{1},\dots$ with $r_{2}\leq n_{1}$, $n_{2}\leq r_{1}+n_{1}$; If (6.1.3) holds for $x_{0},\ x_{1},\dots$ with some $r_{2}$, $n_{2}$, then (6.1.2) holds for $x_{0},\ x_{1},\dots$ with $r_{1}\leq r_{2}+n_{2}$, $n_{1}\leq r_{2}$. Let us consider the $\mathbb{C}_{\infty}$-case. We see that $r_{0}$, resp. $n$ from (1.1.2) are analogs of $r_{2}$, resp. $n_{2}$ of (6.1.3) and of $n_{1}$, resp. $r_{1}$ of (6.1.2). We see that there is no complete analogy between $\mathbb{C}_{\infty}$-case and D-case. Moreover, for the D-case there is no analog of the following fact that holds in the $\mathbb{C}_{\infty}$-case. Let $\goth C$ be the Carlitz module and $x_{0}=0,x_{1},x_{2},\dots$ be a sequence of elements satisfying $x_{i+1}^{q}+\theta x_{i+1}=x_{i}$. It is a holonomic sequence. An analog of the $Exp$ map for the Carlitz module is the ordinary $exp:\mathbb{C}^{+}/2\pi i\to\mathbb{C}^{*}$. Hence, an analog of the above $\{x_{n}\}$ is a sequence $a_{n}:=exp(2\pi i/p^{n})$. It is not holonomic. Nevertheless, analogs of many theorems for the D-case hold for the $\mathbb{C}_{\infty}$-case. For example: Theorem 6.2. Let $x_{0},x_{1},\dots$ and $y_{0},y_{1},\dots$ be holonomic sequences. Then their memberwise sum $x_{0}+y_{0},x_{1}+y_{1},\dots$ is also a holonomic sequence. Proof. Like in [K], [KP], we use a method of indeterminate coefficients, with some minor modifications. For the reader’s convenience, we do not give the general proof, but we consider only some particular cases: the general idea is the same. First, we consider the simplest non-trivial case $r_{0}=n=1$, $\varkappa_{1}=0$ (notations of (1.1)) for both $x_{0},x_{1},\dots$, $y_{0},y_{1},\dots$. For their sum $x_{0}+y_{0},x_{1}+y_{1},\dots$ we have $n=2$, $r_{0}=3$, i.e. there exist $P_{2},\ P_{1},\ P_{0}\in\mathbb{C}_{\infty}[\tau]$ such that $\forall\ i$ we have

[TABLE]

and deg($P_{2})=3$. Moreover, we can choose $P_{1}$ and $P_{0}$ of degrees 2 and 1 respectively. Let us consider a vector space $V$ of dimension 18 over $\mathbb{C}_{\infty}$ whose basis elements are

[TABLE]

(considered as abstract symbols). Conditions that $x_{0},x_{1},\dots$ and $y_{0},y_{1},\dots$ are holonomic sequences give us relations (here $a_{*}=a_{*}(x)$, $b_{*}=b_{*}(x)$ are from (1.1) for $x_{0},\ x_{1},\dots)$:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and analogous relations for $y_{*}$, with coefficients $a_{*}(y)^{q^{*}}$, $b_{*}(y)^{q^{*}}$. There are 10 equalities. We can consider the left hand sides of these equalities as elements of $V$ (linear combinations of basis elements). The matrix of coefficients of these elements is of size $10\times 18$ of rank 10, because $a_{1}(x),\ a_{1}(y)\neq 0$, the matrix consists of two $5\times 9$ diagonal blocks, both of them have a $5\times 5$ triangilar submatrix with the diagonal entries $a_{1}(*)^{q^{*}}$. We denote by $V_{0}$ the subspace of $V$ generated by these elements. Let $V_{1}$ be the subspace of $V$ generated by vectors $x_{i+\alpha}^{q^{\beta}}+y_{i+\alpha}^{q^{\beta}}$ for all admissible combinations of $\alpha,\ \beta$ from (6.2.2). We have dim $V_{1}=9$, dim $V_{0}=10$, hence $V_{0}\cap V_{1}\neq 0$. A non-zero element $v\in V_{0}\cap V_{1}\neq 0$ gives us $P_{2},\ P_{1},\ P_{0}$ from (6.2.1). Really, let

[TABLE]

(the same set of $\alpha,\ \beta$). Then $P_{\alpha}:=\sum_{\beta}c_{\alpha,\beta}\tau^{\beta}$ satisfy (6.2.1). Now, we consider the case when both $x_{0},x_{1},\dots$ and $y_{0},y_{1},\dots$ have $n=1$, the same $r$, and their $\varkappa_{1}$ is $r-1$ (the maximal possible value). We have: $n$ of $\{x_{i}+y_{i}\}$ is 2. Let in (6.2.1) we have deg $P_{2}=\lambda$ (an unknown), deg $P_{1}=\lambda-1$, deg $P_{0}=\lambda-2$. $V$ has a basis

[TABLE]

and the same elements for $y_{i+*}^{q^{*}}$. Hence, we have: dim $V=6\lambda$, dim $V_{1}=3\lambda$. Equations (6.2.3) have the form

[TABLE]

[TABLE]

[TABLE]

($\lambda-r$ equations) and

[TABLE]

[TABLE]

[TABLE]

($\lambda-r+1$ equations). Hence, we have: dim $V_{0}=4\lambda-4r+2$ and the condition $V_{0}\cap V_{1}\neq 0$ is satisfied if $7\lambda-4r+2>6\lambda$. This means that for any $\lambda\geq 4r-1$ there is a non-trivial linear dependence relation between $(x_{i+\alpha}+y_{i+\alpha})^{q^{\beta_{\alpha}}}$, $\alpha=0,\ 1,\ 2$, $\beta_{0}\leq\lambda-2$, $\beta_{1}\leq\lambda-1$, $\beta_{2}\leq\lambda$. For the general case the proof follows the same ideas. One more case is treated below. $\square$ Let $x_{0},x_{1},\dots$ be a holonomic sequence. The set of $P\in\mathbb{C}_{\infty}[T,\tau]$ such that $P(\sum_{i=0}^{\infty}a_{i}T^{i})=0$ is an ideal in $\mathbb{C}_{\infty}[T,\tau]$. By analogy with the D-case we can conjecture that it can be non-principal (compare with [GL21], Proposition 2.3.2 affirming that for a $T$-divisible submodule of $\mathbb{C}_{\infty}[[T]]$ the defining equation is unique). Let us consider an evidence to this fact. 6.3. Let $x_{0},x_{1},\dots$, resp. $y_{0},y_{1},\dots$ be two holonomic sequences satisfied by equations having $n=2$, $r_{0}=r$ arbitrary, $\varkappa_{1},\varkappa_{2}\leq r-1$. Let us find $n$ and $r_{0}$ for their sum $x_{0}+y_{0},\ x_{1}+y_{1},\dots$. Acting as above we choose first $n=4$, $\lambda$ an indeterminate. The analog of (6.2.4) is

[TABLE]

and the same elements for $y_{i+*}^{q^{*}}$. Hence, we have: dim $V=10\lambda-2$, dim $V_{1}=5\lambda-1$. Equations (6.2.3) have the form

[TABLE]

[TABLE]

[TABLE]

[TABLE]

($\lambda-r$ equations)

[TABLE]

[TABLE]

[TABLE]

[TABLE]

($\lambda-r$ equations) and

[TABLE]

[TABLE]

[TABLE]

[TABLE]

($\lambda-r+1$ equations). Hence, we have: dim $V_{0}=6\lambda-6r+2$ and the condition $V_{0}\cap V_{1}\neq 0$ is satisfied if $11\lambda-6r+1>10\lambda-2\ \iff\ \lambda>6r-3$. Now we repeat the above arguments for $n=5$. The analog of (6.2.4) is

[TABLE]

[TABLE]

and the same elements for $y_{i+*}^{q^{*}}$. Hence, we have: dim $V=12\lambda-6$, dim $V_{1}=6\lambda-3$. Analogs of (6.3.1) — (6.3.3) are 4 sets of equations, with $\lambda-r-1$, $\lambda-r$, $\lambda-r$, $\lambda-r+1$ equations in these sets. Hence, we have: dim $V_{0}=8\lambda-8r$ and the condition $V_{0}\cap V_{1}\neq 0$ is satisfied if $14\lambda-8r-3>12\lambda-6\ \iff\ \lambda>4r-\frac{3}{2}$. We see that the ideal of elements $P\in\mathbb{C}_{\infty}[T,\tau]$ such that $P[\sum_{i=0}^{\infty}(x_{i}+y_{i})T^{i}]=0$ contains two elements: the first one has $n=4,\ r_{0}=6r-2$, the second one has $n=5,\ r_{0}=4r-1$. This is an evidence that this ideal is not principal: compare with [KP], Fig. 7.1, p. 143. References [A86] Anderson, G.W. t-motives. Duke Math. J., 1986, vol. 53, No. 2, p. 457 – 502. [E08] Erić, Aleksandra Lj. The resultant of non-commutative polynomials. Mat. Vesnik 60 (2008), no. 1, 3–8. [GGRW05] Gelfand, Israel; Gelfand, Sergei; Retakh, Vladimir; Wilson, Robert Lee. Quasideterminants. Adv. Math. 193 (2005), no. 1, 56 – 141. [G96] Goss, D. Basic structures of function field arithmetic. Springer-Verlag, Berlin, 1996. xiv+422 pp. [GL07] Grishkov A., Logachev, D. Duality of Anderson t-motives. 2007. http://arxiv.org/pdf/math/0711.1928.pdf [GL20] Grishkov A., Logachev, D. Introduction to Anderson t-motives: a survey. 2020. 39 pages. https://arxiv.org/pdf/2008.10657.pdf [GL21] Grishkov A., Logachev, $h^{1}\neq h_{1}$ for Anderson t-motives. J. of Number Theory. 2021, vol. 225, p. 59 – 89. https://arxiv.org/pdf/1807.08675.pdf [HJ20] Hartl U.; Juschka A.-K. Pink’s theory of Hodge structures and the Hodge conjecture over function fields. ”t-motives: Hodge structures, transcendence and other motivic aspects”, Editors G. Böckle, D. Goss, U. Hartl, M. Papanikolas, European Mathematical Society Congress Reports 2020. [K] Kauers, M. The Holonomic Toolkit.

https://www3.risc.jku.at/publications/download etc. [KP] Kauers, Manuel; Paule, Peter. The concrete tetrahedron. Texts Monogr. Symbol. Comput. Springer Wien, New York, Vienna, 2011, x+203 pp. [NP] Namoijam Ch., Papanikolas M. Hyperderivatives of periods and quasi-periods for Anderson t-modules. https://arxiv.org/pdf/2103.05836.pdf [O33.1] Ore, O. On a special class of polynomials. Trans. Amer. Math. Soc. 35 (1933), no. 3, 559 – 584. [O33.2] Ore, O. Theory of non-commutative polynomials, Ann. of Math. 34 (1933), 480 – 508. [Tg95] Taguchi, Yuichiro. A duality for finite t-modules. J. Math. Sci. Univ. Tokyo 2 (1995), no. 3, 563 – 588.