Symmetric Determinantal Representation of Weakly-Skew Circuits

Grenet, Bruno; Kaltofen, Erich L.; Koiran, Pascal; Portier, Natacha

doi:10.4230/LIPIcs.STACS.2011.543

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Symmetric Determinantal Representation of Weakly-Skew Circuits

Authors Bruno Grenet, Erich L. Kaltofen, Pascal Koiran, Natacha Portier

Part of: Volume: 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)
Part of: Series: Leibniz International Proceedings in Informatics (LIPIcs)
Part of: Conference: Symposium on Theoretical Aspects of Computer Science (STACS)
License: Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported license
Publication Date: 2011-03-11

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Document Identifiers

DOI: 10.4230/LIPIcs.STACS.2011.543
URN: urn:nbn:de:0030-drops-30426

Subject Classification

Keywords

algebraic complexity
determinant and permanent of symmetric matrices
formulas
skew circuits
Valiant’s classes

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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.

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Bruno Grenet, Erich L. Kaltofen, Pascal Koiran, and Natacha Portier. Symmetric Determinantal Representation of Weakly-Skew Circuits. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 543-554, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011) https://doi.org/10.4230/LIPIcs.STACS.2011.543

Author Details

Bruno Grenet

Erich L. Kaltofen

Pascal Koiran

Natacha Portier

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