Powerfree values of binary forms
Let \(f\) be a binary form over the integers of degree \(n\) and having no fixed \(k\)-th power divisor. Improving a result used by \textit{F. Gouvêa} and \textit{B. Mazur} in connection with work on elliptic curves [J. Am. Math. Soc. 4, No. 1, 1--23 (1991; Zbl 0725.11027)], a result of the reviewer [Q. J. Math., Oxf. II. Ser. 43, 45--65 (1992; Zbl 0768.11034)] implies that infinitely often \(f(m,n)\) is \(k\)-th powerfree provided \(n\leq 2k+1\), or provided only that \(k\ge 2\) when \(n=6\). The author injects the methods that \textit{M. Huxley} and \textit{M. Nair} [Proc. Lond. Math. Soc. (3) 41, 66--82 (1980; Zbl 0435.10026)] used to obtain improvements in the results of C. Hooley for the corresponding problem involving polynomials in a single variable. The author's principal result applies when \(k\ge (2\sqrt{2}-1) n/4\), but he also obtains a positive result when \(n\le 2k+2\) as soon as \(k\ge 5\).
scientific article; zbMATH DE number 701301