Symmetric Determinantal Representation of Weakly-Skew Circuits

Bruno Grenet, Erich L. Kaltofen, Pascal Koiran, Natacha Portier, Marc Herbstritt
2011 Symposium on Theoretical Aspects of Computer Science  
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
more » ... cteristic 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.
doi:10.4230/lipics.stacs.2011.543 dblp:conf/stacs/GrenetKKP11 fatcat:lkeemh3dnbcjloosj7nigiongu