Density of power-free values of polynomials II

TL;DR

This paper proves that certain polynomials of degree at least 3 take on the expected density of (d-1)-free values, extending previous results for higher degrees using new methods.

Contribution

It establishes the density of (d-1)-free values for a broader class of polynomials of degree at least 3, improving upon earlier degree restrictions.

Findings

Polynomials of degree ≥ 3 have the expected density of (d-1)-free values.

Extension of previous results from degree ≥ 5 to degree ≥ 3.

New methods enable analysis of polynomial values at lower degrees.

Abstract

In this paper we prove that polynomials $F(x_1, \cdots, x_n) \in \mathbb{Z}[x_1, \cdots, x_n]$ of degree $d \geq 3$, satisfying certain hypotheses, take on the expected density of $(d-1)$-free values. This extends the authors' earlier result where a different method implied the similar statement for polynomials of degree $d\geq 5$.