On Continuity of the Roots of a Parametric Zero Dimensional Multivariate Polynomial Ideal
On Continuity of the Roots of a Parametric Zero Dimensional Multivariate Polynomial Ideal
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DOI：
10.1145/3208976.3209004
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发表时间：
2018-07
期刊：
影响因子：
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通讯作者：
Yosuke Sato;Ryoya Fukasaku;Hiroshi Sekigawa
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文献类型：
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作者：
Yosuke Sato;Ryoya Fukasaku;Hiroshi Sekigawa
设F=F1(A，X)，…，fl(A，X)是Q[A，X]中的有限多项式集合，变量A=A1，…，Am，X=X1，…，Xn。我们研究了映射θ从Cm的元素a到Cn的子集的连续性，定义为θ(A)=“多项式理想的零点”。设G=(G1，S1)，…，(Gk，Sk)是以A为参数的综合Gröbner系统。根据综合Gröbner系统的一个基本性质，当理想对某些aın Si是零维时，对任何aın Si也是零维的，并且θ(A)的基数在计算它们的重数时在Si上是相同的.本文证明了θ在Si上也是连续的。我们的结果确保了作者之一最近提出的实量词消除算法的正确性。
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 ". Let G=(G1, S1),..., (Gk, Sk) be a comprehensive Gröbner system of regarding A as parameters. By a basic property of a comprehensive Gröbner system, when the ideal 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.