Square-free values of decomposable forms

TL;DR

This paper proves that certain decomposable forms with integer coefficients take infinitely many square-free values under specific conditions, extending previous results in number theory.

Contribution

It generalizes Greaves' theorem to a broader class of decomposable forms with degree constraints, establishing infinite square-free values.

Findings

Decomposable forms with degree ≤ 2n+2 take infinitely many square-free values.

The result applies under simple necessary conditions.

Generalizes previous work by Greaves.

Abstract

In this paper we prove that decomposable forms, or homogeneous polynomials $F(x_1, \cdots, x_n)$ with integer coefficients which split completely into linear factors over $\mathbb{C}$, take on infinitely many square-free values subject to simple necessary conditions and $\operatorname{deg} f \leq 2n + 2$ for all irreducible factors $f$ of $F$. This work generalizes a theorem of Greaves.