License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2022.119
URN: urn:nbn:de:0030-drops-157158
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/15715/
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van Melkebeek, Dieter ; Morgan, Andrew

Polynomial Identity Testing via Evaluation of Rational Functions

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LIPIcs-ITCS-2022-119.pdf (0.7 MB)

Abstract

We introduce a hitting set generator for Polynomial Identity Testing based on evaluations of low-degree univariate rational functions at abscissas associated with the variables. In spite of the univariate nature, we establish an equivalence up to rescaling with a generator introduced by Shpilka and Volkovich, which has a similar structure but uses multivariate polynomials in the abscissas.
We study the power of the generator by characterizing its vanishing ideal, i.e., the set of polynomials that it fails to hit. Capitalizing on the univariate nature, we develop a small collection of polynomials that jointly produce the vanishing ideal. As corollaries, we obtain tight bounds on the minimum degree, sparseness, and partition size of set-multi-linearity in the vanishing ideal. Inspired by an alternating algebra representation, we develop a structured deterministic membership test for the vanishing ideal. As a proof of concept we rederive known derandomization results based on the generator by Shpilka and Volkovich, and present a new application for read-once oblivious arithmetic branching programs that provably transcends the usual combinatorial techniques.

BibTeX - Entry

@InProceedings{vanmelkebeek_et_al:LIPIcs.ITCS.2022.119,
  author =	{van Melkebeek, Dieter and Morgan, Andrew},
  title =	{{Polynomial Identity Testing via Evaluation of Rational Functions}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{119:1--119:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/15715},
  URN =		{urn:nbn:de:0030-drops-157158},
  doi =		{10.4230/LIPIcs.ITCS.2022.119},
  annote =	{Keywords: Derandomization, Gr\"{o}bner Basis, Lower Bounds, Polynomial Identity Testing}
}

Keywords: Derandomization, Gröbner Basis, Lower Bounds, Polynomial Identity Testing
Collection: 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)
Issue Date: 2022
Date of publication: 25.01.2022

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