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

##### External Resource

https://doi.org/10.4153/S0008414X21000080

(Publisher version)

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##### Fulltext (public)

2005.06713.pdf

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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.

$a_1(\alpha) f_1(\alpha)^n + \ldots + a_k(\alpha) f_k(\alpha)^n$ 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.