An effective algorithm for quantifier elimination over algebraically closed fields using straight line programs

Abstract

In this paper we obtain an effective algorithm for quantifier elimination over algebraically closed fields: For every effective infinite integral domain k, closed under the extraction of pth roots when the characteristic p of k is positive, and every prenex formula ϑ with r blocks of quantifiers involving s polynomials F 1, h., F s ϵ k[ X 1, h., X n ] encoded in dense form, there exists a well-parallelizable algorithm without divisions whose output is a quantifier-free formula equivalent to ϑ. The sequential complexity of this algorithm is bounded by O(¦ϑ¦) + D (O(n)) r , where ¦ϑ¦ is the length of ϑ and D ≥ n is an upper bound for 1 + bE i = 1 s deg F i , and the polynomials in the output are encoded by means of a straight line program. The complexity bound obtained is better than the bounds of the known elimination algorithms, which are of the type ¦ϑ¦.D n cr , where c ≥ 2 is a constant. This becomes notorious when r = 1 (i.e., when there is only one block of quantifiers): the complexity bounds known up to now are not less than D n 2 , while our bound is D O( n) . Moreover, in the particular case that there is only one block of existential quantifiers and the input polynomials are given by a straight line program, we construct an elimination algorithm with even better bounds which depend on the length of this straight line program: Given a formula of the type ∃X n−m+…,∃X n:F 1(X 1,…,X n=0 ∧ … ∧ F s(X 1,…,X n) =0 ∧G 1(X 1,…,X n)≠0 ∧ … ∧ G s ′ (X 1,…,X n)≠0, where F 1, h., F s ϵ k[ X 1, h., X n ] are polynomials whose degrees in the m variables X − m + 1 , h., X n are bounded by an integer d ≥ m and G 1, h., G s′ ϵ k[ X 1, h., X n ] are polynomials whose degrees in the same variables are bounded by an integer δ, this algorithm eliminates quantifiers in time L 2.( s. s′. δ) O(1) . d O( m) , where L is the length of the straight line program that encodes F 1, h., F s , G 1, h., G s′ . Finally, we construct a fast algorithm to compute the Chow Form of an irreducible projective variety. The construction of all the algorithms mentioned above relies on a preprocessing whose cost exceedes the complexity classes considered (they are based on the construction of correct test sequences). In this sense, our algorithms are non-uniform but may be considered uniform as randomized algorithms (choosing the correct test sequences randomly).