On continuity of the roots of a parametric zero dimensional multivariate polynomial ideal

Yosuke Sato, Ryoya Fukasaku, Hiroshi Sekigawa

Research output: Chapter in Book/Report/Conference proceedingConference contribution

2 Citations (Scopus)

Abstract

Let F = ( f1 (Ā,X¯ ), . . ., fl (Ā,X¯ )) be a finite set of polynomials in Q[Ā,X¯ ] with variables Ā = A1, . . .,Am and X¯ = X1, . . .,Xn . We study the continuity of the map θ from an element ā of Cm to a subset of Cn defined by θ (ā) = “the zeros of the polynomial ideal hf1 (ā,X¯ ), . . ., fl (ā,X¯ )i”. Let G = ((G1, S1), . . ., (Gk , Sk )) be a comprehensive Gröbner system of hF i regarding Ā as parameters. By a basic property of a comprehensive Gröbner system, when the ideal hf1 (ā,X¯ ), . . ., fl (ā,X¯ )i is zero dimensional for some ā ∈ Si , it is also zero dimensional for any ā ∈ Si and the cardinality of θ (ā) 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.

Original languageEnglish
Title of host publicationISSAC 2018 - Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation
PublisherAssociation for Computing Machinery
Pages359-365
Number of pages7
ISBN (Electronic)9781450355506
DOIs
Publication statusPublished - Jul 11 2018
Externally publishedYes
Event43rd ACM International Symposium on Symbolic and Algebraic Computation, ISSAC 2018 - New York, United States
Duration: Jul 16 2018Jul 19 2018

Publication series

NameProceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC

Conference

Conference43rd ACM International Symposium on Symbolic and Algebraic Computation, ISSAC 2018
CountryUnited States
CityNew York
Period7/16/187/19/18

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All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

Sato, Y., Fukasaku, R., & Sekigawa, H. (2018). On continuity of the roots of a parametric zero dimensional multivariate polynomial ideal. In ISSAC 2018 - Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation (pp. 359-365). (Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC). Association for Computing Machinery. https://doi.org/10.1145/3208976.3209004