Separation of Multilinear Circuit and Formula Size

An arithmetic circuit or formula is multilinear if the polynomial computed at each of its wires is multilinear. We give an explicit polynomial f (x1, . . . ,xn) with coefficients in {0,1} such that over any field: 1. f can be computed by a polynomial-size multilinear circuit of depth O(log2 n). 2. Any multilinear formula for f is of size nΩ(logn). This gives a superpolynomial gap between multilinear circuit and formula size, and separates multilinear NC1 circuits from multilinear NC2 circuits. ACM Classification: F.2.2, F.1.3, F.1.2, G.2.0 AMS Classification: 68Q15, 68Q17, 68Q25, 94C05