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

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)≠P(y) whenever f(x)≠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(n¹r)/ (where r is the number of distinct primes dividing m) and a lower bound of w(1). Here we show a lower bound of Ω(log n) when m is a product of two primes and Ω(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, using this liberal (and, we argue, more natural) definition very little is known. While the degree is known to be Ω(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