Complexity of Solution of Simultaneous Multivariate Polynomial Equations

TL;DR

This paper introduces a new reduction algorithm for solving multivariate polynomial equations that produces minimal Rabin bases, leading to exponential lower bounds in space complexity and implications for fundamental complexity class separations.

Contribution

The paper presents an original reduction algorithm that computes Rabin bases, demonstrating exponential space lower bounds and providing new insights into the separation of key complexity classes.

Findings

Proposes a reduction algorithm producing Rabin bases for multivariate polynomials.

Establishes exponential lower bounds for space complexity in algebraic proof certification.

Shows that P, NP, and Co-NP are distinct, and that the polynomial hierarchy does not collapse.

Abstract

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.