Symmetric Determinantal Representation of Formulas and Weakly Skew Circuits

Details Symmetric Determinantal Representation of Formulas and Weakly Skew Circuits Symmetric Determinantal Representation of Formulas and Weakly Skew Circuits

2010-07-22

arxiv.org/abs/1007.3804v4

Randomization, Relaxation, and Complexity in Polynomial Equation Solving, Amer. Math. Soc. (Ed.) (2011) 61-96

Computer Science

Computational Complexity

To appear in the AMS Contemporary Mathematics volume on Randomization, Relaxation, and Complexity in Polynomial Equation Solvi

Scientific paper

We deploy algebraic complexity theoretic techniques for constructing symmetric determinantal representations of for00504925mulas 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 B\"urgisser 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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