Arithmetic Circuits, Syntactic Multilinearity, and the Limitations of Skew Formulae

Functions in arithmetic NC are known to have equivalent constant width polynomial degree circuits, but the converse containment is unknown. In a partial answer to this question, we show that syntactic multilinear circuits of constant width and polynomial degree can be depth-reduced, though the resulting circuits need not be syntactic multilinear. We then focus specifically on polynomial-size syntactic multilinear circuits, and study relationships between classes of functions obtained by imposing various resource (width, depth, degree) restrictions on these circuits. Along the way, we obtain a characterisation of NC (and its arithmetic counterparts) in terms of log width restricted planar branching programs. We also study the power of skew formulae, and show that even exponential sums of these are unlikely to suffice to express the determinant function.