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 - Publisher Copyright:
© 2018 Association for Computing Machinery.
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.
KW - Comprehensive Gröbner system
KW - Continuity
KW - Parametric polynomial system
KW - Real quantifier elimination
UR - http://www.scopus.com/inward/record.url?scp=85054934425&partnerID=8YFLogxK
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 -