Symmetric Determinantal Representation of Formulas and Weakly Skew Circuits

TL;DR

This paper introduces smaller symmetric determinantal representations for formulas and weakly skew circuits, advancing algebraic complexity theory and addressing VNP-completeness of the partial permanent in characteristic 2 fields.

Contribution

It provides new, more efficient symmetric determinantal representations for formulas and weakly skew circuits, and resolves a question about the VNP-completeness of the partial permanent in characteristic 2.

Findings

Representations produce smaller matrices than previous methods.

Partial permanent is not VNP-complete in characteristic 2 fields unless the polynomial hierarchy collapses.

Addresses algebraic complexity questions in fields of characteristic 2.

Abstract

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.