On the distribution of polynomial discriminants: totally real case

Abstract

In the paper we study the distribution of the discriminant $D(P)$ of polynomials $P$ from the class ${\mathcal{P}}_n(Q)$ of all integer polynomials of degree $n$ and height at most $Q$. We evaluate the asymptotic number of polynomials $P\in {\mathcal{P}}_n(Q)$ having all the roots real and satisfying the inequality $|D(P)|\le X$ as $Q\to \infty$ and $X/Q^{2n-2}\to 0$.

14 pages

References in corpus (1)

• On the number of integral quadratic polynomials with bounded heights and discriminants