Multi-k-ic depth three circuit lower bound

In a multi-k -ic depth three circuit every variable appears in at most k of the linear polynomials in every product gate of the circuit. This model is a natural generalization of multilinear depth three circuits that allows the formal degree of the circuit to exceed the number of underlying variables (as the formal degree of a multi-k-ic depth three circuit can be kn where n is the number of variables). The problem of proving lower bounds for depth three circuits with high formal degree has gained in importance following a work by Gupta, Kamath, Kayal and Saptharishi [GKKS13a] on depth reduction to high formal degree depth three circuits. In this work, we show an exponential lower bound for multi-k -ic depth three circuits for any arbitrary constant k .