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))k with $$u\in \mathbb {C}$$u∈C, $$g,h\in \mathbb {C}[x]$$g,h∈C[x] and either $$\deg (g)=2$$deg(g)=2, k is even, $$q=k/2$$q=k/2 or $$\deg (g)\le 1$$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 polynomials f having infinitely many shifted power (S-integral) values, and a complete description of superelliptic equations having infinitely many S-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.
Sparse complete sets for coNP: Solution of the P versus NP problem
