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