Complex polynomials and circuit lower bounds for modular counting

AbstractWe study the power of constant-depth circuits containing negation gates, unbounded fan-in AND and OR gates, and a small number of MAJORITY gates. It is easy to show that a depth 2 circuit of sizeO(n) (wheren is the number of inputs) containingO(n) MAJORITY gates can determine whether the sum of the input bits is divisible byk, for any fixedk>1, whereas it is known that this requires exponentialsize circuits if we have no MAJORITY gates. Our main result is that a constant-depth circuit of size $$2^{n^{o(1)} } $$ containingno(1) MAJORITY gates cannot determine if the sum of the input bits is divisible byk; moreover, such a circuit must give the wrong answer on a constant fraction of the inputs. This result was previously known only fork=2. We prove this by obtaining an approximate representation of the behavior of constant-depth circuits by multivariate complex polynomials.