License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2018.10
URN: urn:nbn:de:0030-drops-88786
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/8878/
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Guo, Zeyu ; Saxena, Nitin ; Sinhababu, Amit

Algebraic Dependencies and PSPACE Algorithms in Approximative Complexity

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LIPIcs-CCC-2018-10.pdf (0.6 MB)

Abstract

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).

BibTeX - Entry

@InProceedings{guo_et_al:LIPIcs:2018:8878,
  author =	{Zeyu Guo and Nitin Saxena and Amit Sinhababu},
  title =	{{Algebraic Dependencies and PSPACE Algorithms in Approximative Complexity}},
  booktitle =	{33rd Computational Complexity Conference (CCC 2018)},
  pages =	{10:1--10:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-069-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{102},
  editor =	{Rocco A. Servedio},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2018/8878},
  URN =		{urn:nbn:de:0030-drops-88786},
  doi =		{10.4230/LIPIcs.CCC.2018.10},
  annote =	{Keywords: algebraic dependence, Jacobian, Arthur-Merlin, approximate polynomial, satisfiability, hitting-set, border VP, finite field, PSPACE, EXPSPACE, GCT Cha}
}

Keywords: algebraic dependence, Jacobian, Arthur-Merlin, approximate polynomial, satisfiability, hitting-set, border VP, finite field, PSPACE, EXPSPACE, GCT Cha
Collection: 33rd Computational Complexity Conference (CCC 2018)
Issue Date: 2018
Date of publication: 04.06.2018

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