A Polynomial Variant of Diophantine Triples in Linear Recurrences

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 ${\alpha }_1,\dots ,{\alpha }_k$. 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.

arXiv comment9 pages

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