On the number of monic integer polynomials with given signature

Abstract

In this paper, we show that the number of monic integer polynomials of degree $d \ge 1$ and height at most H which have no real roots is between $c_1H^{d-1/2}$ and $c_2 H^{d-1/2}$, where the constants $c_2>c_1>0$ depend only on d. (Of course, this situation may only occur for d even.) Furthermore, for each integer s satisfying $0 \le s < d/2$ we show that the number of monic integer polynomials of degree d and height at most H which have precisely 2s non-real roots is asymptotic to $\lambda (d,s)H^{d}$ as $H \rightarrow \infty$. The constants $\lambda (d,s)$ are all positive and come from a recent paper of Bertók, Hajdu, and Pethő. They considered a similar question for general (not necessarily monic) integer polynomials and posed this as an open question.