Distribution and Properties of the Critical Values of Random Polynomials With Non-Independent and Non-Identically Distributed Roots

Abstract

This paper considers the pairing between the distribution of the roots and the distribution of the critical values of random polynomials.  The primary model of random polynomial considered in this paper consists of monic polynomials of degree n with a single complex variable z where the roots of the random polynomial are complex valued random variables that are chosen from two independent sequences of iid, complex valued random variables.  The distribution of the random variables from each of the two sequences are different, producing roots of the random polynomial which have non-identical distributions.  Furthermore, both the iid, complex valued random variables from one of the sequences of random variables and the complex conjugates of those random variables are roots of the random polynomial.  Hence, this model of monic random polynomials of degree n has roots that are random variables which are not independent, due to the dependence based on complex conjugates, and the non-identical distributions which arise from the use of the two independent sequences.  This paper also describes the relationship between the roots and critical values of monic random polynomials of degree n where the roots are chosen to be random variables and their complex conjugates where the random variables are from a sequence of iid complex valued random variables.