### 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