We prove that certain instances of the iterated matrix multiplication (IMM) family of polynomials with N variables and degree n require NΩ( √ n) gates when expressed as a homogeneous depth-five ΣΠΣΠΣ arithmetic circuit with the bottom fan-in bounded by N1/2−ε. By a depth-reduction result of Tavenas, this size lower bound is optimal and can be achieved by the weaker class of homogeneous depth-fo...

The notion of homogeneous tensors is discussed. We show that there is a one-to-one correspondence between multivector fields on a manifold M , homogeneous with respect to a vector field ∆ on M , and first-order polydifferential operators on a closed submanifold N of codimension 1 such that ∆ is transversal to N . This correspondence relates the Schouten-Nijenhuis bracket of multivector fields o...

The notion of homogeneous tensors is discussed. We show that there is a one-to-one correspondence between multivector fields on a manifold M , homogeneous with respect to a vector field ∆ on M , and first-order polydifferential operators on a closed submanifold N of codimension 1 such that ∆ is transversal to N . This correspondence relates the Schouten-Nijenhuis bracket of multivector fields o...