### Abstract

Let F = ( f_{1} (Ā,X^{¯} ), . . ., f_{l} (Ā,X^{¯} )) be a finite set of polynomials in Q[Ā,X^{¯} ] with variables Ā = A_{1}, . . .,A_{m} and X^{¯} = X_{1}, . . .,X_{n} . We study the continuity of the map θ from an element ā of C^{m} to a subset of C^{n} defined by θ (ā) = “the zeros of the polynomial ideal hf_{1} (ā,X^{¯} ), . . ., f_{l} (ā,X^{¯} )i”. Let G = ((G_{1}, S_{1}), . . ., (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 hf_{1} (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 |
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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 | |

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 |
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### Conference

Conference | 43rd ACM International Symposium on Symbolic and Algebraic Computation, ISSAC 2018 |
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Country | United States |

City | New York |

Period | 7/16/18 → 7/19/18 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*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.

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

*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

}

TY - GEN

T1 - On continuity of the roots of a parametric zero dimensional multivariate polynomial ideal

AU - Sato, Yosuke

AU - Fukasaku, Ryoya

AU - Sekigawa, Hiroshi

PY - 2018/7/11

Y1 - 2018/7/11

N2 - 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.

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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UR - http://www.scopus.com/inward/citedby.url?scp=85054934425&partnerID=8YFLogxK

U2 - 10.1145/3208976.3209004

DO - 10.1145/3208976.3209004

M3 - Conference contribution

T3 - Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC

SP - 359

EP - 365

BT - ISSAC 2018 - Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation

PB - Association for Computing Machinery

ER -