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Search: MSC category 14R10 ( Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) )

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1. CJM 2014 (vol 67 pp. 286)

Bell, Jason P.; Lagarias, Jeffrey C.
A Skolem-Mahler-Lech Theorem for Iterated Automorphisms of $K$-algebras
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
Categories:11D45, 14R10, 11Y55, 11D88

2. CJM 2003 (vol 55 pp. 533)

Edo, Eric
Automorphismes modérés de l'espace affine
Le probl\`eme de Jung-Nagata ({\it cf.}\ [J], [N]) consiste \`a savoir s'il existe des automorphismes de $k[x,y,z]$ qui ne sont pas mod\'er\'es. Nous proposons une approche nouvelle de cette question, fond\'ee sur l'utilisation de la th\'eorie des automates et du polygone de Newton. Cette approche permet notamment de g\'en\'eraliser de fa\c con significative les r\'esultats de [A]. The Jung-Nagata's problem ({\it cf.}\ [J], [N]) asks if there exists non-tame (or wild) automorphisms of $k[x,y,z]$. We give a new way to attack this question, based on the automata theory and the Newton polygon. This new approch allows us to generalize significantly the results of [A].

Keywords:tame automorphisms, automata, Newton polygon

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