Homogeneous Formulas and Symmetric Polynomials

TLDR: In this paper, it was shown that every multilinear homogeneous formula for symmetric polynomials has size at least O(n 2 ) and that product-depth d multi-inear formulas have size at most 2^{2^{Omega(k^{1/d})}n}

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.