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A Generalised Skolem–Mahler–lech Theorem for Affine Varieties
Author(s) -
Bell Jason P.
Publication year - 2006
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s002461070602268x
Subject(s) - subvariety , mathematics , finite field , affine transformation , set (abstract data type) , automorphism , sequence (biology) , variety (cybernetics) , field (mathematics) , finite set , point (geometry) , discrete mathematics , pure mathematics , combinatorics , mathematical analysis , geometry , statistics , biology , computer science , genetics , programming language
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 σ m ( 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.