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 P n (Q) of all integer polynomials of degree n and height at most Q. We evaluate the asymptotic number of polynomials P ∈ P n (Q) having all the roots real and satisfying the inequality |D(P )| ≤ X as Q → ∞ and X/Q 2n−2 → 0. Brief outlineSection 1 contains the background of the subject and the main results of the paper. Section 2 includes auxiliary proposions necessary to prove the main results; the reader can skip this section in the first reading. In Section 3 we express the distribution function of the discriminant via the volumes of specific regions. In Section 4 we study the asymptotic behaviour of the corresponding volume for small values of discriminant of polynomials having all the roots real.Everywhere in the paper the degree n ≥ 4 of polynomials is fixed; the parameter Q bounding heights of polynomials from above grows to infinity.