Monotone separations for constant degree polynomials

Abstract

We prove a separation between monotone and general arithmetic formulas for polynomials of constant degree. We give an example of a polynomial C in n variables and degree k which is computable by a homogeneous arithmetic formula of size O ( k 2 n 2 ) , but every monotone formula computing C requires size ( n / k c ) Ω ( log k ) , with c ∈ ( 0 , 1 ) . Since the upper bound is achieved by a homogeneous arithmetic formula, we also obtain a separation between monotone and homogeneous formulas, for polynomials of constant degree.