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/sub 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}/sup n/ we have P(x)/spl ne/P(y) whenever f(x)/spl ne/f(y). Barrington, Beigel, and Rudich investigated the minimal degree of a polynomial representing the OR function in this way, proving an upper bound of O(n/sup 1/r)/ (where r is the number of distinct primes dividing m) and a lower bound of w(1). Here we show a lower bound of /spl Omega/(log n) when m is a product of two primes and /spl Omega/(log n)/sup 1/(r-1)/ in general. While many lower bounds are known for a much stronger form of representation of a function by a polynomial, using this liberal (and, we argue, more natural) definition very little is known. While the degree is known to be /spl Omega/(log n) for the generalized inner product because of its high communication complexity, our bound is the best known for any function of low communication complexity and any modulus not a prime power. >