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Arithmetic Circuits and the Hadamard Product of Polynomials

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Abstract

Motivated by the Hadamard product of matrices we define the Hadamard product of multivariate polynomials and study its arithmetic circuit and branching program complexity. We also give applications and connections to polynomial identity testing. Our main results are the following. \begin{itemize} \item[$\bullet$] We show that noncommutative polynomial identity testing for algebraic branching programs over rationals is complete for the logspace counting class $\ceql$, and over fields of characteristic $p$ the problem is in $\ModpL/\Poly$. \item[$\bullet$] We show an exponential lower bound for expressing the Raz-Yehudayoff polynomial as the Hadamard product of two monotone multilinear polynomials. In contrast the Permanent can be expressed as the Hadamard product of two monotone multilinear formulas of quadratic size. \end{itemize}

BibTeX - Entry

@InProceedings{arvind_et_al:LIPIcs:2009:2304,
  author =	{Vikraman Arvind and Pushkar S. Joglekar and Srikanth Srinivasan},
  title =	{{Arithmetic Circuits and the Hadamard Product of Polynomials}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science},
  pages =	{25--36},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-13-2},
  ISSN =	{1868-8969},
  year =	{2009},
  volume =	{4},
  editor =	{Ravi Kannan and K. Narayan Kumar},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2009/2304},
  URN =		{urn:nbn:de:0030-drops-23046},
  doi =		{http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2009.2304},
  annote =	{Keywords: Hadamard product, identity testing, lower bounds, algebraic branching programs}
}

Keywords: Hadamard product, identity testing, lower bounds, algebraic branching programs
Seminar: IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science
Issue date: 2009
Date of publication: 2009


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