Friable values of binary forms (Q439892)

Let \(P^+(n)\) denote the largest prime factor of \(n\) and let \(F\in\mathbb Z[X,Y]\) be an integral binary form of degree \(t\geq 2\). The authors consider the problem of how small \(y\) can be as a function of \(x\) in order that NEWLINE\[NEWLINE\Psi_F(x,y):= \text{card}\{1\leq a,b\leq x: P^+(F(a,b))\leq y\}\times x^2.\tag{\(*\)}NEWLINE\]NEWLINE When \(F= X^2+ Y^2\), \textit{P. Moree} in [Manuscr. Math. 80, No. 2, 199--211 (1993; Zbl 0791.11046)] showed that \((*)\) holds with \(y= x^\varepsilon\). Suppose that \(F\in\mathbb Z[X,Y]\) with degree \(t\geq 2\) has no repeated irreducible factors, where \(k\) of these irreducible factors have the largest degree \(g\) and \(l\) have degree \(g-1\).NEWLINENEWLINE In Theorem 1 the authors prove that for any \(\varepsilon> 0\) \((*)\) holds for \(y\geq x^{\alpha_F+\varepsilon}\), where \(\alpha_F= g-{2\over k}\) if \(k\geq 2\) and, if \(k= 1\), \(\alpha_F= g-1-{1\over l+1}\) or \({2\over 3}\) according as \((g, t)\neq(2,3)\) or \((g, t)= (2, 3)\). When \(F\) is a cubic form Theorem 2 provides an improvement of this result by establishing that \(\alpha_F={1\over \sqrt{e}}\) or \(0\) according as \(F\) is irreducible or reducible. The proofs are intricate and depend on establishing three further propositions as well as utilizing results or methods contained in papers in the long list of references. In particular the proof of Theorem 1 extends the ideas in [Period. Math. Hung. 43, No. 1--2, 111--119 (2001; Zbl 0980.11041)] by \textit{C. Dartyge}, \textit{G. Martin} and \textit{G. Tenenbaum}, where a similar problem for \(F\in\mathbb Z]X]\) was considered.