On the power-free values of polynomials in two variables. II
type: Item
instance of: scholarly article
title: On the power-free values of polynomials in two variables. II
zbMATH Open document ID: 1233.11101
author: Christopher Hooley
published in: Journal of Number Theory
publication date: 2009-05-28T00:00:00Z
review text: Let \(f(x,y)\in\mathbb{Z}[x,y]\) be a polynomial of degree \(r\) which is irreducible over the rationals, and not merely a function of some linear combination \(ax+by\). Then if \(s\geq 2\) is given, one expects that \(f\) attains infinitely many \(s\)-th power-free values \(f(m,n)\), providing that there is no non-trivial \(s\)-th power dividing every value \(f(m,n)\). In the first paper of this series [Analytic number theory. Essays in honour of Klaus Roth on the occasion of his 80th birthday. Cambridge: Cambridge University Press, 235--266 (2009; Zbl 1226.11098)] it was shown that this holds for \(s\geq\tfrac34 r-1\), and indeed that there are \(\gg X^2\) \(s\)-th power-free values for \(m,n\leq X\). When \(f\) splits into linear factors over \(\overline{\mathbb{Q}}\) it was shown that it suffices to have \(s\geq\tfrac12 r-1\). The method failed to yield an asymptotic formula for the number of \(s\)-th power-free values obtained. The present paper achieves such an asymptotic formula in the case in which \(f\) splits over \(\overline{\mathbb{Q}}\), again assuming that \(s\geq\tfrac12 r-1\). The proof builds on ideas from the previous work, which are developed further here. In particular there is a careful analysis of the distribution of the roots of the congruence \(f(x,y)\equiv 0\pmod{p^2}\), at a point where the previous argument used only the distribution modulo \(p\).
Mathematics Subject Classification ID: 11N32
Mathematics Subject Classification ID: 11N25
Mathematics Subject Classification ID: 11N36
zbMATH DE Number: 5559090
zbMATH Keywords: polynomials in 2 variables
zbMATH Keywords: power-free values
zbMATH Keywords: asymptotic formula
reviewed by: D. R. Heath-Brown
MaRDI profile type: Publication
full work available at URL: j.jnt.2008.12.006
OpenAlex ID: W2064949878
cites work: Q4092051
cites work: Q3613913
DOI: 10.1016/J.JNT.2008.12.006
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description: scientific article; zbMATH DE number 5559090