Arithmetic Circuits and the Hadamard Product of Polynomials

TL;DR

This paper introduces the Hadamard product for multivariate polynomials, explores its complexity in arithmetic circuits and branching programs, and presents new results on polynomial identity testing and lower bounds.

Contribution

It defines the Hadamard product for polynomials, analyzes its complexity, and connects it to polynomial identity testing and lower bounds for specific polynomials.

Findings

Identity testing for noncommutative polynomials is complete for $ ext{C}_= ext{L}$ over rationals.

Exponential lower bounds for expressing certain polynomials as Hadamard products.

Permanent can be expressed as a Hadamard product of quadratic-size formulas.

Abstract

Motivated by the Hadamard product of matrices we define the Hadamard product of multivariate polynomials and study its arithmetic circuit and branching program complexity. We also give applications and connections to polynomial identity testing. Our main results are the following. 1. We show that noncommutative polynomial identity testing for algebraic branching programs over rationals is complete for the logspace counting class $\ceql$, and over fields of characteristic $p$ the problem is in $\ModpL/\Poly$. 2.We show an exponential lower bound for expressing the Raz-Yehudayoff polynomial as the Hadamard product of two monotone multilinear polynomials. In contrast the Permanent can be expressed as the Hadamard product of two monotone multilinear formulas of quadratic size.