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arXiv · 2008.10328

Integral zeros of a polynomial with linear recurrences as coefficients

Clemens Fuchs·Sebastian Heintze
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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.

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Clemens Fuchs, Sebastian Heintze. 2020-08-24. Integral zeros of a polynomial with linear recurrences as coefficients. https://doi.org/10.1016/j.indag.2021.03.001

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