Abstract
AbstractWe prove that apart from explicitly given cases, described in terms of Dickson polynomials, a polynomial$$f\in \mathbb {Q}[x]$$f∈Q[x]can have at most one shift$$f(x)-\lambda $$f(x)-λ$$(\lambda \in \mathbb {C})$$(λ∈C)of the form$$u(g(x))^q(h(x))^k$$u(g(x))q(h(x))kwith$$u\in \mathbb {C}$$u∈C,$$g,h\in \mathbb {C}[x]$$g,h∈C[x]and either$$\deg (g)=2$$deg(g)=2,kis even,$$q=k/2$$q=k/2or$$\deg (g)\le 1$$deg(g)≤1,$$k\ge 2$$k≥2,$$q\ge 1$$q≥1. This is shown by handling the case of two possible shifts, which was an open issue. As an application, we give a precise statement yielding a description of polynomialsfhaving infinitely many shifted power (S-integral) values, and a complete description of superelliptic equations having infinitely manyS-integral solutions when the polynomial involved is composite. In the case where there are finitely many solutions, our results yield effective bounds for them. Finally, as further applications, we give effective results for polynomial values in the solutions of Pell equations and in non-degenerate binary recurrence sequences.
Funder
NKFIH
European Union and the European Social Fund
Publisher
Springer Science and Business Media LLC
Subject
Algebra and Number Theory
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