Algebraic dependencies and PSPACE algorithms in approximative complexity

TL;DR

This paper introduces new complexity bounds for algebraic dependence testing of polynomials, showing it lies in AM ∩ coAM, and studies the complexity of approximate polynomial satisfiability, placing it in PSPACE, with implications for algebraic complexity theory.

Contribution

It proves algebraic dependence testing is in AM ∩ coAM and introduces the concept of approximate polynomial satisfiability, placing it in PSPACE, advancing understanding of these problems' complexity.

Findings

Algebraic dependence testing is unlikely to be NP-hard.

Approximate polynomial satisfiability is NP-hard but in PSPACE.

Hitting-set construction for ar VP is in PSPACE.

Abstract

Testing whether a set $\mathbf{f}$ of polynomials has an algebraic dependence is a basic problem with several applications. The polynomials are given as algebraic circuits. Algebraic independence testing question is wide open over finite fields (Dvir, Gabizon, Wigderson, FOCS'07). The best complexity known is NP$^{\#\rm P}$ (Mittmann, Saxena, Scheiblechner, Trans.AMS'14). In this work we put the problem in AM $\cap$ coAM. In particular, dependence testing is unlikely to be NP-hard and joins the league of problems of "intermediate" complexity, eg. graph isomorphism & integer factoring. Our proof method is algebro-geometric-- estimating the size of the image/preimage of the polynomial map $\mathbf{f}$ over the finite field. A gap in this size is utilized in the AM protocols. Next, we study the open question of testing whether every annihilator of $\mathbf{f}$ has zero constant term (Kayal, CCC'09). We give a geometric characterization using Zariski closure of the image of $\mathbf{f}$; introducing a new problem called approximate polynomials satisfiability (APS). We show that APS is NP-hard and, using projective algebraic-geometry ideas, we put APS in PSPACE (prior best was EXPSPACE via Grobner basis computation). As an unexpected application of this to approximative complexity theory we get-- Over any field, hitting-set for $\overline{\rm VP}$ can be designed in PSPACE. This solves an open problem posed in (Mulmuley, FOCS'12, J.AMS 2017); greatly mitigating the GCT Chasm (exponentially in terms of space complexity).