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

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 [x 1 ,...,x 1 ], 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 (c 1 ·l) c r 2 r q , for some constants c 1 , c 2 . If no ∀-quantifiers occur and the matrix of φ is in DNF, then the time is bounded by (c 1 ·l)rc q 2 .