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.
Categories: math.NT