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We consider the problem of finding a condition for a univariate polynomialhaving a given multiplicity structure when the number of distinct roots isgiven. It is well known that such conditions can be written as conjunctions ofseveral polynomial equations and one inequation in the coefficients, by usingrepeated parametric gcd's. In this paper, we give a novel condition which isnot based on repeated gcd's. Furthermore, it is shown that the number ofpolynomials in the condition is optimal and the degree of polynomials issmaller than that in the previous condition based on repeated gcd's.

A Condition for Multiplicity Structure of Univariate Polynomials