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：我们考虑了复杂的根分类问题,即对所有可能的多种结构的复杂根基找到单亚多面系数的条件,众所周知,这种条件可以作为若干多面方程式和系数的一个正数的结合来写,这些系数中的多面方程式被称为多面方程式,众所周知,通过反复的参数格德可以取得对立物。由此产生的对立物通常是嵌套的决定因素,即输入的基数的决定因素,因此也是儿子。在本文中,我们给出了一种新型的对立物,这些不以反复的格德方程式为基础。新的对立物比较简单,因为它们是非排斥性的决定因素,其最高程度较小。