Sums of read-once formulas: How many summands are necessary?

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