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Square-free Values of Decomposable Forms
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- Journal:
- Canadian Journal of Mathematics / Volume 70 / Issue 6 / 01 December 2018
- Published online by Cambridge University Press:
- 20 November 2018, pp. 1390-1415
- Print publication:
- 01 December 2018
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- Article
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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.
