We consider the problem of finding a condition for a univariate polynomial having a given multiplicity structure when the number of distinct roots is given. It is well known that such conditions can be written as conjunctions of several polynomial equations and one inequation in the coefficients, by using repeated parametric gcd's. In this paper, we give a novel condition which is not based on repeated gcd's. Furthermore, it is shown that the number of polynomials in the condition is optimal and the degree of polynomials is smaller than that in the previous condition based on repeated gcd's.

中文翻译：

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一元多项式多重结构的条件

我们考虑在给定不同根数的情况下为具有给定多重结构的单变量多项式找到条件的问题。众所周知，通过使用重复的参数gcd，可以将这样的条件写为几个多项式方程和一个系数不等式的结合。在本文中，我们给出了一种不基于重复gcd的新颖条件。此外，基于重复的gcd，示出了该条件下的多项式的数量是最优的，并且多项式的次数小于先前条件下的多项式的次数。