Title: Deciding freeness of modules over skew polynomial rings



Abstract:

Systems of differential or difference equations over a differential/difference field $K$ can be described by left modules over the skew polynomial ring generated over $K$ by the occurring differential and difference operators, and solutions in certain field extensions $F$ are given by homomorphisms of these modules into the field $F$.

Hence, information on the structures of these modules is quite desirable.

In this talk, we discuss how to check whether such a module is free as a module over the subring generated by some of the operators.



Motivated by questions on t-motives, we will focus on the special case of two difference operators $\pi$ and $\rho$.

We will provide algorithms to check if such modules are finitely generated as modules over $K[\rho]$, and whether they are even free modules.

In the case that the answer is yes, the result of the algorithm readily tells the rank, and provides a basis.



We will conclude the talk by explaining, how this result solves the mentioned questions in the theory of t-motives.