• DOI:10.1090/conm/556
• Corpus ID: 14523508

Symmetric Determinantal Representation of Formulas and Weakly Skew Circuits

@article{Grenet2010SymmetricDR,
  title={Symmetric Determinantal Representation of Formulas and Weakly Skew Circuits},
  author={Bruno Grenet and Erich L. Kaltofen and Pascal Koiran and Natacha Portier},
  journal={ArXiv},
  year={2010},
  volume={abs/1007.3804}
}

• Bruno Grenet, E. Kaltofen, +1 author Natacha Portier
• Published 21 July 2010
• Mathematics
• ArXiv

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 Burgisser 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.