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An efficient quantifier elimination algorithm for algebraically closed fields of any characteristic

Published: 01 November 1975 Publication History

Abstract

An Algorithm is given which, as input, accepts a formula φ in prenex normal form, with prime formulas P=o, P≠o, for P ε Z [x1,...,x1], and some r ε N. It produces a quantifierfree formula ψ in disjunctive normal form (DNF) such that, in any algebraically closed field, φ ψ is true. The time required for a formula of length l with q quantifiers and r-q free variables is bounded by (c1·l) cr2 rq, for some constants c1, c2. If no ∀-quantifiers occur and the matrix of φ is in DNF, then the time is bounded by (c1·l)rcq2.

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  • (2005)Some new effectivity bounds in computational geometryApplied Algebra, Algebraic Algorithms and Error-Correcting Codes10.1007/3-540-51083-4_54(131-151)Online publication date: 1-Jun-2005
  • (2005)Grobner bases, Ritt's algorithm and decision procedures for algebraic theoriesApplied Algebra, Algebraic Algorithms and Error-Correcting Codes10.1007/3-540-51082-6_81(230-237)Online publication date: 1-Jun-2005
  • (2005)Absolute primality of polynomials is decidable in random polynomial time in the number of variablesAutomata, Languages and Programming10.1007/3-540-10843-2_2(16-28)Online publication date: 25-May-2005
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Published In

cover image ACM SIGSAM Bulletin
ACM SIGSAM Bulletin  Volume 9, Issue 4
November 1975
32 pages
ISSN:0163-5824
DOI:10.1145/1088322
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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 November 1975
Published in SIGSAM Volume 9, Issue 4

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Cited By

View all
  • (2005)Some new effectivity bounds in computational geometryApplied Algebra, Algebraic Algorithms and Error-Correcting Codes10.1007/3-540-51083-4_54(131-151)Online publication date: 1-Jun-2005
  • (2005)Grobner bases, Ritt's algorithm and decision procedures for algebraic theoriesApplied Algebra, Algebraic Algorithms and Error-Correcting Codes10.1007/3-540-51082-6_81(230-237)Online publication date: 1-Jun-2005
  • (2005)Absolute primality of polynomials is decidable in random polynomial time in the number of variablesAutomata, Languages and Programming10.1007/3-540-10843-2_2(16-28)Online publication date: 25-May-2005
  • (1998)An effective algorithm for quantifier elimination over algebraically closed fields using straight line programsJournal of Pure and Applied Algebra10.1016/S0022-4049(97)00068-6129:2(173-200)Online publication date: Aug-1998
  • (1997)Polar Varieties, Real Equation Solving, and Data StructuresJournal of Complexity10.1006/jcom.1997.043213:1(5-27)Online publication date: 1-Mar-1997
  • (1990)Precise sequential and parallel complexity bounds for quantifier elimination over algebraically closed fieldsJournal of Pure and Applied Algebra10.1016/0022-4049(90)90159-F67:1(1-14)Online publication date: Nov-1990
  • (1988)A bibliography of quantifier elimination for real closed fieldsJournal of Symbolic Computation10.1016/S0747-7171(88)80016-65:1-2(267-274)Online publication date: 1-Feb-1988
  • (1983)Definability and fast quantifier elimination in algebraically closed fieldsTheoretical Computer Science10.1016/0304-3975(83)90002-624:3(239-277)Online publication date: Aug-1983

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