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A generalised Skolem-Mahler-Lech theorem for affine varieties. (English) Zbl 1147.11020
J. Lond. Math. Soc., II. Ser. 73, No. 2, 367-379 (2006); corrigendum ibid. 78, No. 1, 267-272 (2008).
Let $$K$$ be a field of characteristic $$0$$. The Skolem-Mahler-Lech Theorem states that if $$\{ u_n\}_{n\in\mathbb Z}$$ is a two-sided linear recurrence sequence with entries in $$K$$, then the set of $$n\in\mathbb Z$$ with $$u_n=0$$ is a union of a finite set, and finitely many doubly infinite arithmetic progressions. As the author explains in his paper, this can be restated as follows. If $$\sigma : K^n\to K^n$$ is an invertible linear map, $${\mathbf v}\in K^n$$, and $$W$$ a linear subspace of $$K^n$$ of codimension $$1$$, then the set of $$n\in\mathbb Z$$ with $$\sigma^n({\mathbf v})\in W$$ is a union of a finite set and finitely many doubly infinite arithmetic progressions. His main result is the following generalization:
Let $$Y$$ be an affine variety over $$K$$, $${\mathbf q}\in Y (K)$$, $$\sigma$$ an automorphism of $$Y$$ defined over $$K$$, and $$X$$ a subvariety of $$Y$$ defined over $$K$$. Then the set of $$n\in\mathbb Z$$ with $$\sigma^n({\mathbf q})\in X$$ is a union of a finite set and finitely many doubly infinite arithmetic progressions.
His first theorem proves this in the special case that $$Y$$ is the $$N$$-dimensional affine space $${\mathbf A}^N$$, and there he proceeds in a similar manner as Skolem-Mahler-Lech, by embedding $$K$$ into $$\mathbb Q_p$$ for a suitable prime $$p$$, and then showing that there are a finite number of $$p$$-adic power series $$f_1,\ldots ,f_r\in\mathbb Q_p[[z]]$$ such that every integer $$n$$ under consideration is a zero of one of the $$f_i$$. This is more involved than in the proof of Skolem-Mahler-Lech. Then he reduces the general case of arbitrary affine varieties $$Y$$ to affine spaces by embedding $$Y$$ into $${\mathbf A}^N$$ for an appropriate $$N$$ and lifting $$\sigma$$ to an automorphism $${\mathbf A}^N$$. This is possible by a theorem of V. Srinivas [Math. Ann. 289, 125–132 (1991; Zbl 0725.14003)].

##### MSC:
 11D45 Counting solutions of Diophantine equations 11D88 $$p$$-adic and power series fields 11Y55 Calculation of integer sequences 14R10 Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)
##### Keywords:
Affine varieties; automorphisms; Skolem-Mahler-Lech theorem
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