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

Yosuke Sato; Ryoya Fukasaku; Hiroshi Sekigawa

doi:10.1145/3208976.3209004

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 proceeding › Conference contribution

3 Citations (Scopus)

Abstract

Let F = ( f₁ (Ā,X^¯ ), . . ., f_l (Ā,X^¯ )) be a finite set of polynomials in Q[Ā,X^¯ ] with variables Ā = A₁, . . .,A_m and X^¯ = X₁, . . .,X_n . We study the continuity of the map θ from an element ā of C^m to a subset of Cⁿ defined by θ (ā) = “the zeros of the polynomial ideal hf₁ (ā,X^¯ ), . . ., f_l (ā,X^¯ )i”. Let G = ((G₁, S₁), . . ., (G_k , S_k )) 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^¯ ), . . ., f_l (ā,X^¯ )i is zero dimensional for some ā ∈ S_i , it is also zero dimensional for any ā ∈ S_i and the cardinality of θ (ā) is identical on S_i counting their multiplicities. In this paper, we prove that θ is also continuous on S_i . Our result ensures the correctness of an algorithm for real quantifier elimination one of the authors has recently developed.

Original language	English
Title of host publication	ISSAC 2018 - Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation
Publisher	Association for Computing Machinery
Pages	359-365
Number of pages	7
ISBN (Electronic)	9781450355506
DOIs	https://doi.org/10.1145/3208976.3209004
Publication status	Published - Jul 11 2018
Externally published	Yes
Event	43rd ACM International Symposium on Symbolic and Algebraic Computation, ISSAC 2018 - New York, United States Duration: Jul 16 2018 → Jul 19 2018

Publication series

Name	Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC

Conference

Conference	43rd ACM International Symposium on Symbolic and Algebraic Computation, ISSAC 2018
Country/Territory	United States
City	New York
Period	7/16/18 → 7/19/18

All Science Journal Classification (ASJC) codes

Mathematics(all)

Access to Document

10.1145/3208976.3209004

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

On continuity of the roots of a parametric zero dimensional multivariate polynomial ideal. / Sato, Yosuke; Fukasaku, Ryoya; Sekigawa, Hiroshi.

ISSAC 2018 - Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation. Association for Computing Machinery, 2018. p. 359-365 (Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

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. Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC, Association for Computing Machinery, pp. 359-365, 43rd ACM International Symposium on Symbolic and Algebraic Computation, ISSAC 2018, New York, United States, 7/16/18. https://doi.org/10.1145/3208976.3209004

Sato Y, Fukasaku R, Sekigawa H. 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. Association for Computing Machinery. 2018. p. 359-365. (Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC). doi: 10.1145/3208976.3209004

Sato, Yosuke ; Fukasaku, Ryoya ; Sekigawa, Hiroshi. / On continuity of the roots of a parametric zero dimensional multivariate polynomial ideal. ISSAC 2018 - Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation. Association for Computing Machinery, 2018. pp. 359-365 (Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC).

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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 Gr{\"o}bner system of hF i regarding Ā as parameters. By a basic property of a comprehensive Gr{\"o}bner 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.",

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AB - 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.

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

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