On the Skolem problem and some related questions for parametric families of linear recurrence sequences

Abstract

We show that in a parametric family of linear recurrence sequences a 1 pαqf 1 pαq n `. . . `ak pαqf k pαq n with the coefficients a i and characteristic roots f i , i " 1, . . . , k, given by rational functions over some number field, for all but a set of α of bounded height in the algebraic closure of Q, the Skolem problem is solvable, and the existence of a zero in such a sequence can be effectively decided. We also discuss several related questions.

Skolem’s conjecture confirmed for a family of exponential equations, II

“…Bertók and Hajdu [3,4] proved that in some sense Skolem's conjecture is valid for 'almost all' equations. For strongly related problems and results concerning recurrence sequences, see the papers [10,11,8], and the references there.…”