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.