A Polynomial Variant of Diophantine Triples in Linear Recurrences

TL;DR

This paper investigates polynomial Diophantine triples within linear recurrence sequences, proving finiteness results under certain dominant root conditions and restrictions on polynomial coefficients.

Contribution

It introduces a polynomial variant of Diophantine triples in linear recurrences and establishes finiteness results under specific algebraic conditions.

Findings

Finitely many such triples exist under the dominant root condition.

The result depends on the non-square nature of certain polynomial coefficients.

The work extends classical Diophantine triple results to polynomial recurrence sequences.

Abstract

Let $ (G_n)_{n=0}^{\infty} $ be a polynomial power sum, i.e. a simple linear recurrence sequence of complex polynomials with power sum representation $ G_n = f_1\alpha_1^n + \cdots + f_k\alpha_k^n $ and polynomial characteristic roots $ \alpha_1,\ldots,\alpha_k $. For a fixed polynomial $ p $, we consider triples $ (a,b,c) $ of pairwise distinct non-zero polynomials such that $ ab+p, ac+p, bc+p $ are elements of $ (G_n)_{n=0}^{\infty} $. We will prove that under a suitable dominant root condition there are only finitely many such triples if neither $ f_1 $ nor $ f_1 \alpha_1 $ is a perfect square.