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Square-free Values of Decomposable Forms

Published online by Cambridge University Press:  20 November 2018

Stanley Yao Xiao
Stanley Yao Xiao*
Affiliation:
Mathematical Institute, University of Oxford, Oxford, UK, e-mail: stanley.xiao@maths.ox.ac.uk
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Abstract

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In this paper we prove that decomposable forms, or homogeneous polynomials $F\left( {{x}_{1}}\,,\,.\,.\,.\,,\,{{x}_{n}} \right)$ with integer coefficients that split completely into linear factors over $\mathbb{C}$, take on infinitely many square-free values subject to simple necessary conditions, and they have $\text{deg}\,f\,\le \,2n\,+\mid 2$ for all irreducible factors $f$ of $F$. This work generalizes a theorem of Greaves.

Keywords

11B05square-free valuedecomposable formSelberg sieve
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 70 , Issue 6 , 01 December 2018 , pp. 1390 - 1415
Copyright
Copyright © Canadian Mathematical Society 2018

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