Complexity of Quantifier Elimination in the Theory of Algebraically Closed Fields

An algorithm is described producing for each formula of the first order theory of algebraically closed fields an equivalent free of quantifiers one. Denote by N a number of polynomials occuring in the formula, by d an upper bound on the degrees of polynomials, by n a number of variables, by a a number of quantifier alternations (in the prefix form). Then the algorithm works within the polynomial in the formula's size and in (Nd)n(2a+2) time. Up to now a bound (Nd)no(n) was known ([5], [7], [15]).

[1]  D. Buchsbaum,et al.  Commutative Algebra, Vol. II. , 1959 .

[2]  I. Shafarevich.  Basic algebraic geometry , 1974 .

[3]  George E. Collins,et al.  Quantifier elimination for real closed fields by cylindrical algebraic decomposition , 1975 .

[4]  H. R. Wüthrich,et al.  Ein Entscheidungsverfahren für die Theorie der reell- abgeschlossenen Körper , 1976, Komplexität von Entscheidungsproblemen 1976.

[5]  D. Lazard.  Algèbre linéaire sur $K[X_1,\dots,X_n]$ et élimination , 1977 .

[6]  Dima Grigoriev.  Multiplicative Complexity of a Bilinear Form over a Commutative Ring , 1981, MFCS.

[7]  Daniel Lazard,et al.  Resolution des Systemes d'Equations Algebriques , 1981, Theor. Comput. Sci..

[8]  Erich Kaltofen.  A Polynomial-Time Reduction from Bivariate to Univariate Integral Polynomial Factorization , 1982, FOCS.

[9]  László Lovász,et al.  Factoring polynomials with rational coefficients , 1982 .

[10]  Erich Kaltofen,et al.  A polynomial reduction from multivariate to bivariate integral polynomial factorization. , 1982, STOC '82.

[11]  Erich Kaltofen,et al.  A polynomial-time reduction from bivariate to univariate integral polynomial factorization , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[12]  Arjen K. Lenstra,et al.  Factoring multivariate polynomials over finite fields , 1983, J. Comput. Syst. Sci..

[13]  Joos Heintz,et al.  Corrigendum: Definability and Fast Quantifier Elimination in Algebraically Closed Fields , 1983, Theor. Comput. Sci..