Symmetric determinantal representation of formulas and weakly skew circuits

Abstract

Abstract. We deploy algebraic complexity theoretic techniques to construct 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.

A Short Proof of the Symmetric Determinantal Representation of Polynomials

“…In [11BG], B. Grenet et al gives a proof of the HMV Theorem using a construction that, from the very beginning, was meant to be valid for any field of characteristic different from 2. As they point out (see [11BG,Sec.…”

“…One of the merits of our proof that makes it short and elementary is that it requires no prior knowledge of multidimensional systems theory (as compared to [06HMV] and [12RQ]) or advanced representation theory for multivariate polynomials (as compared to [11BG]). For instance, the proof of the HMV theorem in both [06HMV] and [12RQ], it is necessary to know or first prove that an arbitrary real polynomial has a "linear description."…”

“…An alternative proof of the HMV Theorem was given by B. Grenet et al in [11BG] which is based on symmetrizing the algebraic complexity theoretic construction by L. Valiant in [79LV]. More precisely, it is proved in [79LV] that every polynomial has a nonsymmetric determinantal representation which is proved using a weighted digraph construction and then in [11BG] they use a similar yet symmetrized construction to get a symmetric determinantal representation to prove the HMV Theorem.…”

“…An alternative proof of the HMV Theorem was given by B. Grenet et al in [11BG] which is based on symmetrizing the algebraic complexity theoretic construction by L. Valiant in [79LV]. More precisely, it is proved in [79LV] that every polynomial has a nonsymmetric determinantal representation which is proved using a weighted digraph construction and then in [11BG] they use a similar yet symmetrized construction to get a symmetric determinantal representation to prove the HMV Theorem. As such, the proof in [11BG] requires a bit of effort to derive the HMV Theorem as there is quite a bit of prior knowledge needed to do this construction, especially if you are not familiar with algebraic complexity theory.…”

Algebraic Complexity Classes

“…The converse is also almost true. As shown by Grenet, Kaltofen, Koiran and Portier in [GKKP11], over any field of characteristic other than 2, Det n is a projection of SymDet n 3 . Characteristic 2 is a problem: symmetric matrices correspond to undirected graphs, so each undirected cycle gives rise to two directed cycles, and so to get a projection we need division by 2.…”