A condition for multiplicity structure of univariate polynomials (Q2229732)

The paper under review studies the problem of finding conditions for a given parametric univariate polynomial having a given multiplicity structure when the number of distinct roots is given as well. For example, let us consider the quatric parametric polynomial \(F:=a_4x^4+a_3x^3+a_2x^2+a_1x+a_0\) where \(a_i\)'s are the parameters taking their values in the field of complex numbers. The main question addressed in this paper is the following: Assume that we are given a multiplicity structure, e.g. \((3,1)\). Then, we are looking for conditions on the \(a_i\)'s such that \(F\) has two distinct complex roots, say \(r_1\) and \(r_2\), where the multiplicities of these roots are \(3\) and \(1\), respectively. The classical method for this purpose is based on using repeated parametric gcd's. In this paper, the authors give a novel method by introducing a certain generalization of Sylvester resultant of \(F\) and \(F'\) to obtain only a single polynomial condition with degree smaller than those in the previous method.