Linear equations on $t$-modules

Let $F$ be a number field.

Given finitely many $F$-valued points on a commutative algebraic group defined over $F$, a question of interest to number theorists is the determination of the group of their linear relations.

In this article, we investigate an analogous problem in the $t$-module setting.

Let $L$ be a global function field, and $E$ be a $d$-dimensional $t$-module defined over $L$.

Given finitely many points on $E$ with entries in $L$, we establish the connection between their $\mathbb{F}_q[t]$-linear relations and polynomial solutions of Frobenius difference equations.

Consequently, we deduce an algorithm to compute the module of their $\mathbb{F}_q[t]$-linear relations.