Lower bounds on representing Boolean functions as polynomials in Zm

The MOD_m-degree of Boolean function F is defined to be the smallest degree of any polynomial P, over the ring of integers modulo m, such that for all 0-1 assignments x, F(x)=0 iff P(x)=0. By exploring the periodic property of the binomial coefficients module m, two new lower bounds on the MOD_m-degree of the MOD_l and not-MOD_m functions are proved, where m is any composite integer and l has a prime factor not dividing m. Both bounds improve from n^Ω(1) in D.A.M. Barrington et al. (1992) to Ω(n). A lower bound, n/2, for the majority function and a lower bound, √n, for the MidBit function are also proved