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# A Skolem-Mahler-Lech Theorem for Iterated Automorphisms of $K$-algebras

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Published:2014-02-10
Printed: Apr 2015
• Jason P. Bell,
Department of Pure Mathematics , University of Waterloo , Waterloo, ON , N2L 3G1
• Jeffrey C. Lagarias,
Department of Mathematics , University of Michigan , Ann Arbor, MI 48109-1043 , USA
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## Abstract

This paper proves a commutative algebraic extension of a generalized Skolem-Mahler-Lech theorem due to the first author. Let $A$ be a finitely generated commutative $K$-algebra over a field of characteristic $0$, and let $\sigma$ be a $K$-algebra automorphism of $A$. Given ideals $I$ and $J$ of $A$, we show that the set $S$ of integers $m$ such that $\sigma^m(I) \supseteq J$ is a finite union of complete doubly infinite arithmetic progressions in $m$, up to the addition of a finite set. Alternatively, this result states that for an affine scheme $X$ of finite type over $K$, an automorphism $\sigma \in \operatorname{Aut}_K(X)$, and $Y$ and $Z$ any two closed subschemes of $X$, the set of integers $m$ with $\sigma^m(Z ) \subseteq Y$ is as above. The paper presents examples showing that this result may fail to hold if the affine scheme $X$ is not of finite type, or if $X$ is of finite type but the field $K$ has positive characteristic.
 Keywords: automorphisms, endomorphisms, affine space, commutative algebras, Skolem-Mahler-Lech theorem
 MSC Classifications: 11D45 - Counting solutions of Diophantine equations 14R10 - Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) 11Y55 - Calculation of integer sequences 11D88 - $p$-adic and power series fields

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