Diophantine Equations for Polynomial Recursive Sequences

TL;DR

This paper investigates Diophantine equations involving polynomial power sums over number fields, establishing conditions for infinite solutions and analyzing polynomial decompositions in linear recurrence sequences.

Contribution

It applies the Bilu-Tichy criterion to show bounded denominators for solutions and studies polynomial decompositions in second and third order linear recurrences.

Findings

Equation $U_n(x)=V_m(y)$ has infinitely many solutions with bounded $ ext{O}_S$-denominator.

Degree of $g$ in $W_n(x)=g(h(x))$ is bounded for third order linear recurrences.

Degree of $g$ is bounded when $(u_{n_1}+u_{n_2})(x)=g(h(x))$ in binary recurrence sequences.

Abstract

We study the Diophantine equation of type $U_n(x)=V_m(y)$, where $(U_n)_{n\geq 0}$ and $(V_m)_{m\geq 0}$ are polynomial power sums defined over a number field $K$. By applying the finiteness criterion of Bilu and Tichy, we show under appropriate assumptions that equation $U_n(x)=V_m(y)$ has infinitely many solutions with bounded $\mathcal{O}_S$-denominator. We also study decomposable polynomials in third and second order linear recurrence sequences. In particular, we show that if $W_n(x)=g(h(x))$ for a simple third order linear recurrence sequence $(W_n(x))_{n\geq 0}$ of complex polynomials, then deg $g$ is bounded. Furthermore, we show that if $(u_{n_1}+u_{n_2})(x)=g(h(x))$ for a binary recurrence sequence $(u_n(x))_{n\geq 0}$ then deg $g$ is bounded.