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 equations and one inequation in the coefficients. Those polynomials in the coefficients are called discriminants for multiplicities. It is well known that discriminants can be obtained by using repeated parametric gcd's. The resulting discriminants are usually nested determinants, that is, determinants of matrices whose entries are determinants, and so son. In this paper, we give a new type of discriminants which are not based on repeated gcd's. The new discriminants are simpler in that they are non-nested determinants and have smaller maximum degrees.

翻译：我们考虑复数根分类问题，即寻找单变量多项式系数满足其复数根所有可能重数结构的条件。已知这类条件可表示为系数域上若干多项式方程与一个不等式方程的合取形式。这些关于系数的多项式被称为重数判别式。现有方法通常通过重复参数化最大公因式（GCD）获取判别式，所得结果通常是嵌套行列式——即矩阵元素本身为行列式的分层结构。本文提出一类不基于重复GCD的新型判别式。这类新判别式具有非嵌套的行列式结构与更小的最高阶数，因而更为简洁。