Real and complex pseudozero sets for polynomials with applications

¹ Laboratoire LIP6, département Calcul scientifique, université Pierre et Marie Curie, 4 place Jussieu, F-75252, Paris Cedex 05, France; stef.graillat@lip6.fr
² DALI Project, Laboratory LP2A, université de Perpignan, 52, avenue Paul Alduy, 66860 Perpignan Cedex, France; langlois@univ-perp.fr

Abstract

Pseudozeros are useful to describe how perturbations of polynomial coefficients affect its zeros. We compare two types of pseudozero sets: the complex and the real pseudozero sets. These sets differ with respect to the type of perturbations. The first set – complex perturbations of a complex polynomial – has been intensively studied while the second one – real perturbations of a real polynomial – seems to have received little attention. We present a computable formula for the real pseudozero set and a comparison between these two pseudozero sets. We conclude that the complex pseudozero sets have to be preferred except when the perturbed real polynomials admit non-real zeros. We also give some applications of pseudozero set in control theory.