Complexity of Solution of Simultaneous Multivariate Polynomial Equations [article]

Duggirala Meher Krishna, Duggirala Ravi
<span title="2021-05-23">2021</span> <span class="release-stage" >pre-print</span>
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
more &raquo; ... ves 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. It is also shown that the class of languages decidable by 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 and NP. It follows, from the discussions, that the complexity classes NP and P are not the same, by relativization of the said probabilistic polynomial time complexity class, with respect to P and NP. The only main open question that is not answered in this paper is as to whether or not Co-NP and NP are the same.
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