Lower bounds for approximations by low degree polynomials over Z/sub m/

We use a Ramsey-theoretic argument to obtain the first lower bounds for approximations over Z/sub m/ by nonlinear polynomials: (i) A degree-2 polynomial over Z/sub m/ (m odd) must differ from the parity function on at least a 1/2-1/2((log n)/sup /spl Omega/(1)/) fraction of all points in the Boolean n-cube. A degree-O(1) polynomial over Z/sub 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/sub m/oAND/sub O(1)/ circuits (i.e., circuits with a single majority-gate at the output node, MOD/sub m/-gates at the middle level, and constant-fanin AND-gates at the input level) that compute parity: (i) MAJoMOD/sub m/oAND/sub 2/ circuits that compute parity must have top fanin 2((log n)/sup /spl Omega/(1)/). (ii) Parity cannot be computed by MAJoMODmoAND/sub O(1)/ circuits with top fanin O(1). Similar results hold for the MOD/sub q/ function as well.