On Continuity of the Roots of a Parametric Zero Dimensional Multivariate Polynomial Ideal

Abstract

Let F= f1(A, X),...,fl(A, X) be a finite set of polynomials in Q[A, X] with variables A=A1,...,Am and X=X1,...,Xn. We study the continuity of the map θ from an element a of Cm to a subset of Cn defined by θ(a)= " the zeros of the polynomial ideal < f1(a, X),..., fl(a, X) >". Let G=(G1, S1),..., (Gk, Sk) be a comprehensive Gröbner system of < F > regarding A as parameters. By a basic property of a comprehensive Gröbner system, when the ideal < f1(a, X),..., fl(a, X) > is zero dimensional for some a ın Si, it is also zero dimensional for any a ın Si and the cardinality of θ(a) is identical on Si counting their multiplicities. In this paper, we prove that θ is also continuous on Si. Our result ensures the correctness of an algorithm for real quantifier elimination one of the authors has recently developed.