Read-once polynomials: How many summands suffice?

TL;DR

This paper investigates the minimal number of read-once formulas needed to express certain multilinear polynomials, establishing tight lower bounds for these representations.

Contribution

It provides the first tight lower bounds on the number of summands required in sum-of-read-once formulas for specific multilinear polynomials.

Findings

Established tight lower bounds for the number of summands.

Identified classes of multilinear polynomials requiring many ROFs.

Enhanced understanding of the complexity of polynomial representations.

Abstract

An arithmetic read-once formula (ROF) is a formula (circuit of fan-out 1) over $+, \times$ where each variable labels at most one leaf. Every multilinear polynomial can be expressed as the sum of ROFs. In this work, we prove, for certain multilinear polynomials, a tight lower bound on the number of summands in such an expression.