On the power-free values of polynomials in two variables. II
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 \(p\).
scientific article; zbMATH DE number 5559090