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)].

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)].

Reviewer: Jan-Hendrik Evertse (Leiden)