Effective results for hyper- and superelliptic equations over number fields

TL;DR

This paper establishes explicit bounds on solutions to hyper- and superelliptic equations over number fields, providing criteria for the non-existence of solutions when the exponent exceeds a certain explicit bound.

Contribution

It offers fully explicit upper bounds for solutions of hyper- and superelliptic equations over number fields, extending previous results with precise quantitative measures.

Findings

Explicit height bounds for solutions in terms of polynomial and field data

A bound C beyond which no solutions exist unless trivial or roots of unity

Application to equations over arbitrary finitely generated domains

Abstract

We consider hyper- and superelliptic equations $f(x)=by^m$ with unknowns x,y from the ring of S-integers of a given number field K. Here, f is a polynomial with S-integral coefficients of degree n with non-zero discriminant and b is a non-zero S-integer. Assuming that n>2 if m=2 or n>1 if m>2, we give completely explicit upper bounds for the heights of the solutions x,y in terms of the heights of f and b, the discriminant of K, and the norms of the prime ideals in S. Further, we give a completely explicit bound C such that $f(x)=by^m$ has no solutions in S-integers x,y if m>C, except if y is 0 or a root of unity. We will apply these results in another paper where we consider hyper- and superelliptic equations with unknowns taken from an arbitrary finitely generated integral domain of characteristic 0.