Homogeneous formulas and symmetric polynomials

TL;DR

This paper analyzes the complexity of computing elementary symmetric polynomials with various formulas, establishing lower bounds, separations, and constructions that advance understanding of formula size and structure.

Contribution

It provides new lower bounds for multilinear homogeneous formulas, demonstrates superpolynomial separations between formula types, and constructs efficient formulas for symmetric polynomials.

Findings

Lower bounds for multilinear homogeneous formulas of S(k,n)

Superpolynomial separation between multilinear and homogeneous formulas

Efficient homogeneous formulas for S(k,n) matching previous questions

Abstract

We investigate the arithmetic formula complexity of the elementary symmetric polynomials S(k,n). We show that every multilinear homogeneous formula computing S(k,n) has size at least k^(Omega(log k))n, and that product-depth d multilinear homogeneous formulas for S(k,n) have size at least 2^(Omega(k^{1/d}))n. Since S(n,2n) has a multilinear formula of size O(n^2), we obtain a superpolynomial separation between multilinear and multilinear homogeneous formulas. We also show that S(k,n) can be computed by homogeneous formulas of size k^(O(log k))n, answering a question of Nisan and Wigderson. Finally, we present a superpolynomial separation between monotone and non-monotone formulas in the noncommutative setting, answering a question of Nisan.