Skolem-Mahler-Lech type theorems and Picard-Vessiot theory

TL;DR

This paper explores the connections between Skolem-Mahler-Lech type theorems, Picard-Vessiot theory, and the dynamical Mordell-Lang conjecture, providing solutions in specific cases for linear difference equations with rational coefficients.

Contribution

It establishes the equivalence of three key problems in difference equations, Picard-Vessiot extensions, and dynamical systems, and solves two of these problems in a general special case.

Findings

Equivalent formulations of three major problems in difference equations and algebraic dynamics.

Existence of Picard-Vessiot extensions inside the ring of sequences in certain cases.

Solutions to generalized Skolem-Mahler-Lech problems for rational function coefficients.

Abstract

We show that three problems involving linear difference equations with rational function coefficients are essentially equivalent. The first problem is the generalization of the classical Skolem-Mahler-Lech theorem to rational function coefficients. The second problem is the question whether or not for a given linear difference equation there exists a Picard-Vessiot extension inside the ring of sequences. The third problem is a certain special case of the dynamical Mordell-Lang conjecture. This allows us to deduce solutions to the first two problems in a particular but fairly general special case.