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

N1 - Funding Information:
This work was partially supported by JSPS KAKENHI Grant Numbers 17K12642, 18K03426, and 18K11172.

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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U2 - 10.1145/3208976.3209004

DO - 10.1145/3208976.3209004

M3 - Conference contribution

AN - SCOPUS:85054934425

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

T2 - 43rd ACM International Symposium on Symbolic and Algebraic Computation, ISSAC 2018

Y2 - 16 July 2018 through 19 July 2018

ER -