Power-free values of the polynomial t_1...t_r - 1

TL;DR

This paper establishes an asymptotic formula for counting k-free values of the polynomial t_1...t_r - 1 over integer points in a multidimensional box, extending understanding of polynomial value distributions.

Contribution

It provides the first asymptotic count for k-free values of the polynomial t_1...t_r - 1 over integral points, generalizing previous results to multiple variables.

Findings

Asymptotic formula derived for k-free values

Results applicable for polynomial t_1...t_r - 1

Extends previous work on polynomial value distribution

Abstract

Let k, r > 1 be two integers. We prove an asymptotic formula for the number of k-free values of the r variables polynomial t_1...t_r - 1 over the integral points of [1, x]^r.