Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH scholarly article en Grenet, Bruno; Kaltofen, Erich L.; Koiran, Pascal; Portier, Natacha http://www.dagstuhl.de/lipics License
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Symmetric Determinantal Representation of Weakly-Skew Circuits

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Abstract

We deploy algebraic complexity theoretic techniques for constructing symmetric determinantal representations of weakly-skew circuits, which include formulas. Our representations produce matrices of much smaller dimensions than those given in the convex geometry literature when applied to polynomials having a concise representation (as a sum of monomials, or more generally as an arithmetic formula or a weakly-skew circuit). These representations are valid in any field of characteristic different from 2. In characteristic 2 we are led to an almost complete solution to a question of Buergisser on the VNP-completeness of the partial permanent. In particular, we show that the partial permanent cannot be VNP-complete in a finite field of characteristic 2 unless the polynomial hierarchy collapses.

BibTeX - Entry

@InProceedings{grenet_et_al:LIPIcs:2011:3042,
  author =	{Bruno Grenet and Erich L. Kaltofen and Pascal Koiran and Natacha Portier},
  title =	{{Symmetric Determinantal Representation of Weakly-Skew Circuits}},
  booktitle =	{28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011) },
  pages =	{543--554},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-25-5},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{9},
  editor =	{Thomas Schwentick and Christoph D{\"u}rr},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2011/3042},
  URN =		{urn:nbn:de:0030-drops-30426},
  doi =		{http://dx.doi.org/10.4230/LIPIcs.STACS.2011.543},
  annote =	{Keywords: algebraic complexity, determinant and permanent of symmetric matrices, formulas, skew circuits, Valiantís classes}
}

Keywords: algebraic complexity, determinant and permanent of symmetric matrices, formulas, skew circuits, Valiantís classes
Seminar: 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)
Issue date: 2011
Date of publication: 2011


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