On Hall's conjecture

TL;DR

This paper demonstrates the existence of specific polynomial pairs with integer coefficients that satisfy particular degree conditions related to Hall's conjecture, advancing understanding of polynomial degree relationships.

Contribution

It constructs explicit examples of polynomials with prescribed degrees that challenge existing bounds related to Hall's conjecture.

Findings

Existence of polynomials with degrees satisfying the conjecture's conditions

Explicit construction of polynomial examples with degree constraints

Progress towards understanding polynomial degree bounds in number theory

Abstract

We show that for any even positive integer d there exist polynomials x and y with integer coefficients such that deg(x) = 2d, deg(y) = 3d and deg(x^3 - y^2) = d + 5.