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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单变量多项式的重数结构的一个条件

我们考虑在给定不同根的数量时为具有给定重数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，这些条件可以写成几个多项式方程和系数中的一个不等式的结合。在本文中，我们给出了一个不基于重复 gcd 的新条件。此外，还表明该条件中的多项式数量是最优的，多项式的次数小于基于重复gcd的先前条件中的多项式次数。