No abstract

A lower bound is established on degrees of Positivstellensatz calculus refutations (over a real field) introduced in (Grigoriev & Vorobjov 2001; Grigoriev 2001) for the knapsack problem. The bound depends on the values of coefficients of an instance of the knapsack problem: for certain values the lower bound is linear and for certain values the upper bound is constant, while in the polynomial calculus the degree is always linear (regardless of the values of coefficients) (Impagliazzo et al. 1999).This shows that the Positivstellensatz calculus can be strictly stronger than the polynomial calculus from the point of view of the complexity of the proofs.

An algorithm is described producing for each formula of the first order theory of algebraically closed fields an equivalent free of quantifiera one. Denote by N a number of polynomials occuring in the formula, by J., an upper bound on the degrees of polynomials, by n a number of variables, bya, a number of quantifier alternati ona (in the prefix form). Then the algorithm w Î rks within the poly nomial in the formula' s si ze and in ( N cl, ) 11, (ta.+ ) time. Up t o now a bound ( Nd.,)11, 0'(11,

We prove the first exponential lower bound on the size of any depth 3 arithmetic circuit with unbounded fanin computing an explicit function (the determinant) over an arbitrary finite field. This answers an open problem of [N91] and [NW951 for the cs~e of finite fields. We intepret here arithmetic circuits in the algebra of polynomials over the given field. The proof method involves a new argument on the rank of linear functions, and a group symmetry on polynomials vanishing at certain nonsingular matrices, and could be of independent interest.

We introduce two versions of proof systems dealing with systems of inequalities: Positivstellensatz refutations and Positivstellensatz calculus. For both systems we prove the lower bounds on degrees and lengths of derivations for the example due to Lazard, Mora and Philippon. These bounds are sharp, as well as they are for the Nullstellensatz refutations and for the polynomial calculus. The bounds demonstrate a gap between the Null-and Positivstellensatz refutations on one hand, and the polynomial calculus and Positivstellensatz calculus on the other.