Symmetric Determinantal Representation of Formulas and Weakly Skew Circuits - Archive ouverte HAL Access content directly
Book Sections Year : 2011

Symmetric Determinantal Representation of Formulas and Weakly Skew Circuits

(1, 2) , (3) , (1, 2) , (1, 2)
1
2
3

Abstract

We deploy algebraic complexity theoretic techniques for constructing symmetric determinantal representations of formulas and weakly skew circuits. 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 Bürgisser 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.
Fichier principal
Vignette du fichier
preprint111024.pdf (461.66 Ko) Télécharger le fichier
Origin : Files produced by the author(s)

Dates and versions

ensl-00504925 , version 1 (21-07-2010)
ensl-00504925 , version 2 (26-07-2010)
ensl-00504925 , version 3 (13-01-2011)
ensl-00504925 , version 4 (24-10-2011)

Identifiers

Cite

Bruno Grenet, Erich Kaltofen, Pascal Koiran, Natacha Portier. Symmetric Determinantal Representation of Formulas and Weakly Skew Circuits. Leonid Gurvits, Philippe Pebay, J. Maurice Rojas, David Thompson. Randomization, Relaxation, and Complexity in Polynomial Equation Solving, Amer. Math. Soc., pp.61-96, 2011, Contemporary Mathematics, 978-0-8218-5228-6. ⟨10.1090/conm/556⟩. ⟨ensl-00504925v4⟩
147 View
231 Download

Altmetric

Share

Gmail Facebook Twitter LinkedIn More