On the distribution of polynomials with bounded roots, I. Polynomials with real coefficients

Abstract

Let v_d^(s) denote the set of coefficient vectors of contractive polynomials of degree d with 2s non-real zeros. We prove that v_d^(s) can be computed by a multiple integral, which is related to the Selberg integral and its generalizations. We show that the boundary of the above set is the union of finitely many algebraic surfaces. We investigate arithmetical properties of v_d^(s) and prove among others that they are rational numbers. We will show that within contractive polynomials, the ‘probability’ of picking a totally real polynomial decreases rapidly when its degree becomes large.