Integral zeros of a polynomial with linear recurrences as coefficients

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.

13 pages

Cited by in corpus (1)

• Approximation of values of algebraic elements over the ring of power sums