A polynomial variant of diophantine triples in linear recurrences

Clemens Fuchs; Sebastian Heintze

doi:10.1007/s10998-022-00460-y

A polynomial variant of diophantine triples in linear recurrences

Clemens Fuchs^*, Sebastian Heintze

^*Corresponding author for this work

Institute of Analysis and Number Theory (5010)

Research output: Contribution to journal › Article › peer-review

Abstract

Let (Gn)n=0∞ be a polynomial power sum, i.e. a simple linear recurrence sequence of complex polynomials with power sum representation Gn=f1α1n+⋯+fkαkn and polynomial characteristic roots α₁, … , α_k. For a fixed polynomial p, we consider sets { a, b, c} consisting of three non-zero polynomials such that ab+ p, ac+ p, bc+ p are elements of (Gn)n=0∞. We will prove that under a suitable dominant root condition there are only finitely many such sets if neither f₁ nor f₁α₁ is a perfect square.

Original language	English
Pages (from-to)	289-299
Number of pages	11
Journal	Periodica Mathematica Hungarica
Volume	86
Issue number	1
DOIs	https://doi.org/10.1007/s10998-022-00460-y
Publication status	Published - 2023

Keywords

Diophantine triples
Function fields
Linear recurrence sequences

ASJC Scopus subject areas

General Mathematics

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10.1007/s10998-022-00460-yLicence: CC BY 4.0

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title = "A polynomial variant of diophantine triples in linear recurrences",

abstract = "Let (Gn)n=0∞ be a polynomial power sum, i.e. a simple linear recurrence sequence of complex polynomials with power sum representation Gn=f1α1n+⋯+fkαkn and polynomial characteristic roots α1, … , αk. For a fixed polynomial p, we consider sets { a, b, c} consisting of three non-zero polynomials such that ab+ p, ac+ p, bc+ p are elements of (Gn)n=0∞. We will prove that under a suitable dominant root condition there are only finitely many such sets if neither f1 nor f1α1 is a perfect square.",

keywords = "Diophantine triples, Function fields, Linear recurrence sequences",

author = "Clemens Fuchs and Sebastian Heintze",

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AU - Heintze, Sebastian

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N2 - Let (Gn)n=0∞ be a polynomial power sum, i.e. a simple linear recurrence sequence of complex polynomials with power sum representation Gn=f1α1n+⋯+fkαkn and polynomial characteristic roots α1, … , αk. For a fixed polynomial p, we consider sets { a, b, c} consisting of three non-zero polynomials such that ab+ p, ac+ p, bc+ p are elements of (Gn)n=0∞. We will prove that under a suitable dominant root condition there are only finitely many such sets if neither f1 nor f1α1 is a perfect square.

AB - Let (Gn)n=0∞ be a polynomial power sum, i.e. a simple linear recurrence sequence of complex polynomials with power sum representation Gn=f1α1n+⋯+fkαkn and polynomial characteristic roots α1, … , αk. For a fixed polynomial p, we consider sets { a, b, c} consisting of three non-zero polynomials such that ab+ p, ac+ p, bc+ p are elements of (Gn)n=0∞. We will prove that under a suitable dominant root condition there are only finitely many such sets if neither f1 nor f1α1 is a perfect square.

KW - Diophantine triples

KW - Function fields

KW - Linear recurrence sequences

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JF - Periodica Mathematica Hungarica

IS - 1

ER -

A polynomial variant of diophantine triples in linear recurrences

Abstract

Keywords

ASJC Scopus subject areas

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Correction to: A polynomial variant of diophantine triples in linear recurrences (Periodica Mathematica Hungarica, (2023), 86, 1, (289-299), 10.1007/s10998-022-00460-y)

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