Diophantine Equations for Polynomial Recursive Sequences

Darsana N; Sudhansu Sekhar Rout

paper

Diophantine Equations for Polynomial Recursive Sequences

arXiv:2512.20384

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 $O_{S}$ -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.

24 pages