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Quantifier elimination in the theory of an algebraically-closed field

Author:

D. IerardiAuthors Info & Claims

STOC '89: Proceedings of the twenty-first annual ACM symposium on Theory of computing

Pages 138 - 147

https://doi.org/10.1145/73007.73020

Published: 01 February 1989 Publication History

Abstract

In this paper we develop a fast parallel procedure for deciding when a set of multivariate polynomials with coefficients in an arbitrary field K have a common algebraic solution. Moreover, since the proposed algorithm is algebraic, it easily yields a procedure for quantifier elimination in the theory of an arbitrary algebraically closed field.

More precisely, we show how to decide whether m polynomials in n variables, each of degree at most d, with coefficients in an arbitrary field K have a common zero in the algebraic closure of K, using sequential time (mn)^Ω(n)d^Ω(n)²), or parallel time Ω(n³ log³ d log m) with (mn)^Ω(n)d^Ω(n)²) processors, in the operations of the coefficient field K. Using randomization, this may be improved to (mn)^Ω(1)d^Ω(n) time.

In addition, the construction is used give a direct EXPSPACE algorithm for quantifier elimination in the theory of an algebraically-closed field, which runs in PSPACE or parallel polynomial time when restricted to formulas with a fixed number of alternations of quantifiers.

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We prove formally that the first order theory of algebraically closed fields enjoys quantifier elimination, and hence is decidable. This proof is organized in two modular parts. We first reify the first order theory of rings and prove that quantifier ...

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cover image ACM Conferences

STOC '89: Proceedings of the twenty-first annual ACM symposium on Theory of computing

February 1989

600 pages

ISBN:0897913078

DOI:10.1145/73007

Editor:
D. S. Johnson

Copyright © 1989 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 01 February 1989

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STOC89: 21st Annual ACM Symposium on the Theory of Computing

May 14 - 17, 1989

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STOC '89 Paper Acceptance Rate 56 of 196 submissions, 29%;

Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

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Bhargava VSaraf SVolkovich IKhuller SVassilevska Williams V(2021)Reconstruction algorithms for low-rank tensors and depth-3 multilinear circuitsProceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing10.1145/3406325.3451096(809-822)Online publication date: 15-Jun-2021
https://dl.acm.org/doi/10.1145/3406325.3451096
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https://doi.org/10.1109/ICIP.2013.6738283
Grenet BKoiran PPortier N(2013)On the complexity of the multivariate resultantJournal of Complexity10.1016/j.jco.2012.10.00129:2(142-157)Online publication date: 1-Apr-2013
https://dl.acm.org/doi/10.1016/j.jco.2012.10.001
Grenet BKoiran PPortier N(2010)The multivariate resultant is NP-hard in any characteristicProceedings of the 35th international conference on Mathematical foundations of computer science10.5555/1885577.1885619(477-488)Online publication date: 23-Aug-2010
https://dl.acm.org/doi/10.5555/1885577.1885619
Grenet BKoiran PPortier N(2010)The Multivariate Resultant Is NP-hard in Any CharacteristicMathematical Foundations of Computer Science 201010.1007/978-3-642-15155-2_42(477-488)Online publication date: 2010
https://doi.org/10.1007/978-3-642-15155-2_42
Delahaye DMayero M(2006)Quantifier Elimination over Algebraically Closed Fields in a Proof Assistant using a Computer Algebra SystemElectronic Notes in Theoretical Computer Science (ENTCS)10.1016/j.entcs.2005.11.023151:1(57-73)Online publication date: 1-Mar-2006
https://dl.acm.org/doi/10.1016/j.entcs.2005.11.023
Buss JFrandsen GShallit J(2005)The computational complexity of some problems of linear algebraSTACS 9710.1007/BFb0023480(451-462)Online publication date: 10-Jun-2005
https://doi.org/10.1007/BFb0023480
Dantsin EEiter TGottlob GVoronkov A(2001)Complexity and expressive power of logic programmingACM Computing Surveys10.1145/502807.50281033:3(374-425)Online publication date: 1-Sep-2001
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Adleman LHuang M(2001)Counting Points on Curves and Abelian Varieties Over Finite FieldsJournal of Symbolic Computation10.1006/jsco.2001.047032:3(171-189)Online publication date: 1-Sep-2001
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