A Polynomial Variant of Diophantine Triples in Linear Recurrences

Let $(G_n{)}_{n=0}^{\mathrm{\infty }}$ 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 $andpolynomialcharacteristicroots$ \alpha_1,\ldots,\alpha_k $\mathrm{.}Forafixedpolynomial$ p $,weconsidertriples$ (a,b,c) $ofpairwisedistinctnon-zeropolynomialssuchthat$ ab+p, ac+p, bc+p $areelementsof$ (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.