We introduce the polynomial coefficient matrix and identify maximum rank of this matrix under variable substitution as a complexity measure for multivariate polynomials. We use our techniques to prove super-polynomial lower bounds against several classes of non-multilinear arithmetic circuits. In particular, we obtain the following results :

As our main result, we prove that any homogeneous depth-3 circuit for computing the product of d matrices of dimension nn requires (nd−12d) size. This improves the lower bounds by Nisan and Wigderson(1995) when d=(1).

There is an explicit polynomial on n variables and degree at most 2n for which any depth-3 circuit C of product dimension at most n10 (dimension of the space of affine forms feeding into each product gate) requires size 2(n). This generalizes the lower bounds against diagonal circuits proved by Saxena(2007). Diagonal circuits are of product dimension 1.

We prove a n(logn) lower bound on the size of product-sparse formulas. By definition, any multilinear formula is a product-sparse formula. Thus, our result extends the known super-polynomial lower bounds on the size of multilinear formulas by Raz(2006).

We prove a 2(n) lower bound on the size of partitioned arithmetic branching programs. This result extends the known exponential lower bound on the size of ordered arithmetic branching programs given by Jansen(2008).