A Generalised Skolem–Mahler–lech Theorem for Affine Varieties

By Jason P. Bell. 2005

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The Skolem–Mahler–Lech theorem states that if f(n) is a sequence given by a linear recurrence over a field of characteristic 0, then the set of m such that f(m) is equal to 0 is the union of a finite number of arithmetic progressions in m ⩾ 0 and a finite set. We prove that if X is a subvariety of an affine variety Y over a field of characteristic 0 and q is a point in Y, and σ is an automorphism of Y, then the set of m such that σm(q) lies in X is a union of a finite number of complete doubly‐infinite arithmetic progressions and a finite set. We show that this is a generalisation of the Skolem–Mahler–Lech theorem.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a14e28b77b148c3f351936f109820a3a88950037

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Citations in the corpus

Periodic points, linearizing maps, and the dynamical Mordell–Lang problem (2008, methodology): “for some nonzero elements ui. By [ Bel06 , Lemma 3:1] (see also [Lec53]), there is a prime p such that the eld of fractions of R embeds into Qp in such a way that all of the ui, all of the eigenvalues of , and all of the reciprocals of the eigenvalues of are sent to elements of Zp. Note that such a ”
Current trends and open problems in arithmetic dynamics (2018, unknown): “1 (Dynamical Mordell–Lang Conjecture: Bell, 2006, [17]; Denis, 1994, [56]; Ghioca–Tucker, 2009, [86]).”
A case of the dynamical Mordell–Lang conjecture (2012, methodology): “In the cases of endomorphisms of semiabelian varieties (see [11,12,15,34]), of automorphisms of affine varieties (see [5,9]), or of étale endomorphisms in general (see [6]), the underlying maps have no ramification.”
The dynamical Mordell-Lang problem for étale maps (2008, methodology): “The technique used in [Bel06], [BGKT] and [GT09] is a modiﬁcation of a method ﬁrst used by Skolem [Sko34] (and later extended by Mahler [Mah35] and Lech [Lec53]) to treat linear recurrence sequences.”