Complexity of Solution of Simultaneous Multivariate Polynomial Equations

In this paper, an original reduction algorithm for solving simultaneous multivariate polynomial equations is presented. The exponential in complexity, but the well-known algorithms, such as extended Euclidean and Buchberger's algorithm, are superexponential. superexponential complexity of algorithms due to their not being "minimal" a certain sense. produces Groebner basis. proposed achieves required task, via computation determinants parametric Sylvester matrices, Rabin basis, which shown be minimal, when two polynomials reduced at time. minimality basis allows us prove lower bounds space algebraic proof certification, specific computational problem class PSPACE, showing that classes PSPACE P cannot same. By same reasoning, it follows Co-NP either NP P, time hierarchy does collapse. It also BPP languages decidable by bounded error probabilistic with (probabilistic) proofs membership input words any one Co-NP. again, from discussions, same, relativization BPP, respect NP.