Lower Bounds for Approximations by Low Degree Polynomials Over Z_m

• Authors: Noga Alon;Richard Beigel
• Venue: CCC '01 Proceedings of the 16th Annual Conference on Computational Complexity
• Year: 2001

Abstract: We use a Ramsey-theoretic argument to obtain the first lower bounds for approximations over Z_m by nonlinear polynomials: 1) 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)^\Omega(1)} fraction of all points in the Boolean n-cube. 2) 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 MAJ \circ MOD_m \circ AND_{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: 1) MAJ \circ MOD_m \circ AND_2 circuits that compute parity must have top fanin 2^{(log n)^\Omega(1)}. 2) Parity cannot be computed by MAJ \circ MOD_m \circ AND_{O(1)} circuits with top fanin O(1). Similar results hold for the MOD_q function as well.