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

Part of: Sequences and sets Diophantine equations

Published online by Cambridge University Press:  08 February 2021

Alina Ostafe  and
Igor E. Shparlinski
Alina Ostafe*
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, Australia, NSW 2052 e-mail: igor.shparlinski@unsw.edu.au
Igor E. Shparlinski
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, Australia, NSW 2052 e-mail: igor.shparlinski@unsw.edu.au
*
e-mail: alina.ostafe@unsw.edu.au
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Abstract

We show that in a parametric family of linear recurrence sequences $a_1(\alpha ) f_1(\alpha )^n + \cdots + a_k(\alpha ) f_k(\alpha )^n$ 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 elements $\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.

Keywords

Skolem problemperfect powerlinear recurrence sequence

MSC classification

Primary: 11B37: Recurrences 11D61: Exponential equations

Information

Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 3 , June 2022 , pp. 773 - 792
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Copyright
© Canadian Mathematical Society 2021

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