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.