LIPIcs.STACS.2011.543.pdf
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Symmetric Determinantal Representation of Weakly-Skew Circuits
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28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
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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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
https://doi.org/10.4230/LIPIcs.STACS.2011.543
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.
Keywords
- algebraic complexity
- determinant and permanent of symmetric matrices
- formulas
- skew circuits
- Valiant’s classes
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