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

Yosuke Sato, Ryoya Fukasaku, Hiroshi Sekigawa
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引用次数: 6

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.
参数零维多元多项式理想根的连续性
设F= f1(A, X),…,fl(A, X)是Q[A, X]中多项式的有限集合,变量A=A1,…,Am and X=X1,…,Xn。我们研究了从Cm的元素a到Cn的子集的映射θ的连续性,其定义为θ(a)=“多项式理想< f1(a, X),…, fl(a, X) > ' '。设G=(G1, S1),…, (Gk, Sk)是以a为参数的< F >的综合Gröbner系统。利用综合Gröbner系统的基本性质,当理想< f1(a, X)时,…, fl(a, X) >对于某些a ın Si是零维的,对于任何a ın Si也是零维的θ(a)的基数在Si上是相同的,计算它们的多重性。本文证明了θ在Si上也是连续的。我们的结果保证了作者最近开发的一种消除实量词算法的正确性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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