Algebraic Dependencies and PSPACE Algorithms in Approximative Complexity

Authors Zeyu Guo, Nitin Saxena, Amit Sinhababu

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Zeyu Guo
  • Department of Computer Science & Engineering, Indian Institute of Technology Kanpur
Nitin Saxena
  • Department of Computer Science & Engineering, Indian Institute of Technology Kanpur
Amit Sinhababu
  • Department of Computer Science & Engineering, Indian Institute of Technology Kanpur

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Zeyu Guo, Nitin Saxena, and Amit Sinhababu. Algebraic Dependencies and PSPACE Algorithms in Approximative Complexity. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 10:1-10:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


Testing whether a set 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). Previously, the best complexity known was NP^{#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 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 f has zero constant term (Kayal, CCC'09). We give a geometric characterization using Zariski closure of the image of 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 Gröbner basis computation). As an unexpected application of this to approximative complexity theory we get- over any field, hitting-sets for overline{VP} can be verified 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).

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Theory of computation → Complexity classes
  • Mathematics of computing → Computations on polynomials
  • Mathematics of computing → Computations in finite fields
  • algebraic dependence
  • Jacobian
  • Arthur-Merlin
  • approximate polynomial
  • satisfiability
  • hitting-set
  • border VP
  • finite field
  • GCT Chasm


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