Power-free values of binary forms and the global determinant method

TL;DR

This paper improves estimates on the density of k-free values of binary forms and enhances related theorems using a generalized global determinant method, advancing understanding of power-free values in number theory.

Contribution

It introduces a generalized global determinant method to better estimate the distribution of power-free values of binary forms and improves existing theorems on their occurrence.

Findings

Enhanced bounds for the density of k-free values

Improved estimates for the number of power-free values in intervals

Generalization of the global determinant method

Abstract

We give an improved estimate for the density of $k$-free values of integral binary forms with no fixed $k$-th power divisor. Further, we give the corresponding improvement to a theorem of Stewart and Top on the number of power-free values in an interval that may be assumed by a binary form. The approach we use involves a generalization of the global determinant method of Salberger.