Sums of read-once formulas: 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 read-once formulas required to sum to specific multilinear polynomials.

Findings

Established tight lower bounds for sums of read-once formulas

Identified classes of multilinear polynomials requiring many summands

Advances understanding of formula complexity in algebraic computation

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.