# Journal of the European Mathematical Society

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**Volume 17, Issue 3, 2015, pp. 523–533**

**DOI: 10.4171/JEMS/509**

Published online: 2015-03-10

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

Michael Wibmer^{[1]}(1) RWTH Aachen, Germany

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 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.

*Keywords: *Linear difference equations, Picard–Vessiot theory, Skolem–Mahler–Lech theorem, dynamical Mordell–Lang conjecture

Wibmer Michael: Skolem–Mahler–Lech type theorems and Picard–Vessiot theory. *J. Eur. Math. Soc.* 17 (2015), 523-533. doi: 10.4171/JEMS/509