Homogeneous Formulas and Symmetric Polynomials

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.