Parametric “non-nested” discriminants for multiplicities of univariate polynomials

We consider the problem of complex root classification, i.e., finding the conditions on the coefficients of a univariate polynomial for all possible multiplicity structures on its complex roots. It is well known that such conditions can be written as conjunctions of several polynomial equalities and one inequality in the coefficients. Those polynomials in the coefficients are called discriminants for multiplicities. It is also known that discriminants can be obtained using repeated parametric greatest common divisors. The resulting discriminants are usually nested determinants, i.e., determinants of matrices whose entries are determinants, and so on. In this paper, we give a new type of discriminant that is not based on repeated greatest common divisors. The new discriminants are simpler in the sense that they are non-nested determinants and have smaller maximum degrees.