On quadratic fields generated by polynomials
Let \(f(X)\) be an integral polynomial without repeated roots, and write \(S_f(M,N)\) for the number of distinct quadratic fields \(\mathbb{Q}(\sqrt {f(n)})\) for \(M<n\leq M+N\). It was shown by \textit{P. Cutter, A. Granville} and \textit{Thomas J. Tucker} [Can. Math. Bull. 46, 71--79 (2003; Zbl 1044.11087)] that \(S_f(0,N)\sim c_fN\) for polynomials of degree at most 2, with \(c_f=1\) for quadratic polynomials. Moreover they showed that \(S_f(0,N)\sim N\) for every degree at least 2, under the ABC-Conjecture. This paper considers unconditional bounds, by examining the number \(Q_f(s;M,N)\) of integers \(n\in(M,N+M]\) for which \(\mathbb{Q} (\sqrt{f(n)})= \mathbb{Q}(\sqrt s)\). It is shown that \[ Q_f(s;M,N)\ll_fN^{1/2}\log N \tag{*} \] whence \(S_f(M,N)\gg_fN^{1/2}(\log N)^{-1}\). This result is proved using the reviewer's `square-sieve' [Math. Ann. 266, 251--259 (1984; Zbl 0514.10038)], along with various bounds for incomplete character sums. It is natural to ask whether a better result for \(S_f(M,N)\) might have been obtained by estimating \(\sum_sQ_f (s;M,N)^2\) directly. Here one would want to apply the square sieve to the multi-set \(\{f(n_1)f(n_2):M< n_1,n_2\leq M+N\}\). The authors present bounds for certain averages of \(Q_f(s;M,N)\) which improve on (*) in mean, for appropriate ranges of \(s\).
scientific article; zbMATH DE number 5500877