Zero sets of bivariate Hermite polynomials

Abstract

We establish various properties for the zero sets of three families of bivariate Hermite polynomials. Special emphasis is given to those bivariate orthogonal polynomials introduced by Hermite by means of a Rodrigues type formula related to a general positive definite quadratic form. For this family we prove that the zero set of the polynomial of total degree n + m consists of exactly n + m disjoint branches and possesses n + m asymptotes. A natural extension of the notion of interlacing is introduced and it is proved that the zero sets of the family under discussion obey this property. The results show that the properties of the zero sets, considered as affine algebraic curves in R-2, are completely different for the three families analyzed. (c) 2014 Elsevier Inc. All rights reserved.