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

Ostafe, Alina; Shparlinski, Igor

doi:10.4153/S0008414X21000080

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

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Ostafe, Alina
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.4153/S0008414X21000080
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Citation

Ostafe, A., & Shparlinski, I. (2022). On the Skolem problem and some related questions for parametric families of linear recurrence sequences. Canadian Journal of Mathematics, 74(3), 773-792. doi:10.4153/S0008414X21000080.

Cite as: https://hdl.handle.net/21.11116/0000-000A-2211-C

Abstract

We show that in a parametric family of linear recurrence sequences
$a_1(\alpha) f_1(\alpha)^n + \ldots + a_k(\alpha) f_k(\alpha)^n$ with the
coefficients $a_i$ and characteristic roots $f_i$, $i=1, \ldots,k$, given by
rational functions over some number field, for all but a set of $\alpha$ of
bounded height in the algebraic closure of $\mathbb Q$, 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.