Abstract
We study Diophantine equations of type f(x)=g(y), where both f and g have at least two distinct critical points (roots of the derivative) and equal critical values at at most two distinct critical points. Various classical families of polynomials (f_n)_n are such that f_n satisfies these assumptions for all n. Our results cover and generalize several results in the literature on the finiteness of integral solutions to such equations. In doing so, we analyse the properties of the monodromy groups of such polynomials. We show that if f has coefficients in a field K of characteristic zero, and at least two distinct critical points and all distinct critical values, then the monodromy group of f is a doubly transitive permutation group. In particular, f cannot be represented as a composition of lower degree polynomials. Several authors have studied monodromy groups of polynomials with some similar properties. We further show that if f has at least two distinct critical points and equal critical values at at most two of them, and if f(x)=g(h(x)) with g, hin K[x] and deg g>1, then either deg hle 2, or f is of special type. In the latter case, in particular, f has no three simple critical points, nor five distinct critical points.
Highlights
Diophantine equations of type f (x) = g(y) have been of long-standing interest to number theorists
To the proof of Theorem 1.1, in Sect. 3 we show that if K is a field of characteristic zero and f ∈ K [x] has at least two distinct critical points and all distinct critical values, Mon( f ) is a doubly transitive permutation group
Lemma 3.3 follows from the fact that a permutation group of degree at least three is doubly transitive on X if and only if the stabilizer of any x0 ∈ X acts transitively on X \{x0}, see [7, Thm. 4.13]
Summary
Theorem 1.5 Let K be a number field, S a finite set of places of K that contains all Archimedean places, OS the ring of S-integers of K and f, g ∈ K [x] such that deg f ≥ 3, deg g ≥ 3 and deg f < deg g If both f and g have at least two distinct critical points and equal critical values at at most two distinct critical points, and their derivatives do not satisfy (1.3) nor (1.4), the equation f (x) = g(y) has only finitely many solutions with a bounded OS-denominator, unless either (deg f, deg g) = (3, 5), or f is indecomposable and g(x) = f (ν(x)) for some quadratic ν ∈ K [x]. We remark that Theorem 1.1 and Theorem 1.5 are ineffective since they rely on the main result of [5], which in turn relies on Siegel’s classical theorem on integral points on curves, which is ineffective
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