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：我们研究复数根分类问题，即寻找一元多项式系数满足其复数根所有可能重数结构的条件。众所周知，此类条件可表示为系数域上多个多项式方程与一个不等式构成的合取式，其中这些关于系数的多项式称为重数判别式。经典方法通过重复参数化最大公因式得到判别式，所得结果通常为嵌套行列式（即矩阵元素仍为行列式的迭代结构）。本文提出一类新型判别式，其构造不依赖重复最大公因式过程。新型判别式具有非嵌套行列式结构且最高次数更小，因此更为简洁。