Integral zeros of a polynomial with linear recurrences as coefficients

TL;DR

This paper investigates the integral solutions of polynomials with coefficients defined by linear recurrences over number fields, providing a description of zeros and a Hilbert irreducibility-like result under certain conditions.

Contribution

It offers a new characterization of zeros of polynomials with recurrence-based coefficients over number fields, extending classical results to this setting.

Findings

Description of zeros (n,z) in natural numbers and S-integers

Results analogous to Hilbert irreducibility for these polynomials

Conditions under which the zeros can be explicitly described

Abstract

Let $ K $ be a number field, $ S $ a finite set of places of $ K $, and $ \mathcal{O}_S $ be the ring of $ S $-integers. Moreover, let $$ G_n^{(0)} Z^d + \cdots + G_n^{(d-1)} Z + G_n^{(d)} $$ be a polynomial in $ Z $ having simple linear recurrences of integers evaluated at $ n $ as coefficients. Assuming some technical conditions we give a description of the zeros $ (n,z) \in \mathbb{N} \times \mathcal{O}_S $ of the above polynomial. We also give a result in the spirit of Hilbert irreducibility for such polynomials.