Discriminants of fields generated by polynomials of given height (Q6561659)

For \(n, H \in \mathbb N\) let \(\mathcal P_n(H)\) denote the set of all monic polynomials in \(\mathbb Z[X]\) of degree \(n\) with absolute value of its coefficients less than \(H\), and \(\mathcal I_n(H)\) its subset of irreducible polynomials. For \(f \in \mathcal I_n(H)\) let \(\Delta(f)\) denote the discriminant of the number field generated by a root of \(f\), and for \(\Delta \in \mathbb N\) put\N\[\NN_n(H, \Delta) = \# \{f \in \mathcal I_n(H) \mid \Delta(f) = \Delta \}.\N\]\NTheorems 1.1 and 1.2 give upper bounds, uniformly over \(\Delta\), for the number \(N_n(H, \Delta)\) and its summatoric function \(M_n(H, D) = \sum_{|\Delta| \le D} N_n(H, \Delta)\). The bounds of Theorem 1.1 are obtained by combining two results: the determinant method, as previously used by the first author [Bull. Lond. Math. Soc. 45, No. 3, 453--462 (2013; Zbl 1362.11092)] and the square sieve of \textit{D. R. Heath-Brown} [Math. Ann. 266, 251--259 (1984; Zbl 0514.10038)].\N\NThese results are applied to the special case of trinomials, which yields lower bounds for the number of quadratic number fields generated by the square root of the discriminants of trinomials of bounded discriminant. This improves former results of the third author [Proc. Am. Math. Soc. 138, No. 1, 125--132 (2010; Zbl 1218.11102)].\N\NFinally, an error in the above cited paper of the first author is pointed out and its proof adjusted.