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# Square-free Values of Decomposable Forms

Published:2018-02-14
Printed: Dec 2018
• Stanley Yao Xiao,
Mathematical Institute, University of Oxford, Oxford, UK
 Format: LaTeX MathJax PDF

## 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 $\deg f \leq 2n + 2$ for all irreducible factors $f$ of $F$. This work generalizes a theorem of Greaves.
 Keywords: square-free value, decomposable form, Selberg sieve
 MSC Classifications: 11B05 - Density, gaps, topology

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