A lower bound on the mod 6 degree of the OR function

Abstract

We examine the computational power of modular counting, where the modulus m is not a prime power, in the setting of polynomials in Boolean variables over Z m . In particular, we say that a polynomial P weakly represents a Boolean function f (both have n variables) if for any inputs x and y in {0,1}n, we have $P(x)\neq P(y)$ whenever $f(x)\neq f(y)$ . Barrington et al. (1994) investigated the minimal degree of a polynomial representing the OR function in this way, proving an upper bound of O(n 1/ r ) (where r is the number of distinct primes dividing m) and a lower bound of $\omega (1)$ . Here, we show a lower bound of $\Omega (\log n)$ when m is a product of two primes and $\Omega ((\log n)^{1/(r-1)})$ in general. While many lower bounds are known for a much stronger form of representation of a function by a polynomial (Barrington et al. 1994, Tsai 1996), very little is known using this liberal (and, we argue, more natural) definition. While the degree is known to be $\Omega (\log n)$ for the generalized inner product because of its high communication complexity (Grolmusz 1995), our bound is the best known for any function of low communication complexity and any modulus not a prime power.