Extrema of polynomials with real roots and Diophantine equations

Hajdu, Lajos and Herendi, Orsolya (2022) Extrema of polynomials with real roots and Diophantine equations. JOURNAL OF NUMBER THEORY. ISSN 0022-314X (In Press)

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Abstract

There are many results in the literature concerning polynomial values and (shifted) power values of polynomials with consecutive integer roots, or more generally, with roots forming an arithmetic progression. It is an interesting question that how far one can ‘disturb’ the structure of the roots such that the finiteness results still remain valid. Also there are many results into this direction, with adding or removing one or more terms (roots). In this paper we study a case where (part of) the symmetric root structure is preserved, however, we allow (possibly large) increasing gaps between the roots. We prove that the finiteness of the solutions can also be guaranteed under these generalized circumstances. In our proofs we combine Baker's method and the Bilu-Tichy theorem with a new result providing an increasing property of the extremal values of polynomials with distinct real roots satisfying certain symmetry and increasing gap properties.

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