Lower bounds for approximations by low degree polynomials over Z m

We use a Ramsey-theoretic argument to obtain the first lower bounds for approximations over Z_m by nonlinear polynomials: (i) A degree-2 polynomial over Z_m (m odd) must differ from the parity function on at least a 1/2-1/2((log n)^Ω(1)) fraction of all points in the Boolean n-cube. A degree-O(1) polynomial over Z_m (m odd) must differ from the parity function on at least a 1/2-o(1) fraction of all points in the Boolean n-cube. These nonapproximability results imply the first known lower bounds on the top fanin of MAJoMOD_moAND_O(1) circuits (i.e., circuits with a single majority-gate at the output node, MOD_m-gates at the middle level, and constant-fanin AND-gates at the input level) that compute parity: (i) MAJoMOD_moAND₂ circuits that compute parity must have top fanin 2((log n)^Ω(1)). (ii) Parity cannot be computed by MAJoMODmoAND_O(1) circuits with top fanin O(1). Similar results hold for the MOD_q function as well