Identity Testing for constant-width, and commutative, read-once

We give improved hitting-sets for two special cases of Read-once Oblivious Arithmetic Branching Programs (ROABP). First is the case of an ROABP with known variable order. The best hitting-set known for this case had cost (nw)O(logn) where n is the number of variables and w is the width of the ROABP. Even for a constant-width ROABP, nothing better than a quasi-polynomial bound was known. We improve the hitting-set complexity for the known-order case to nO(logw). In particular, this gives the first polynomial time hitting-set for constant-width ROABP (known-order). However, our hitting-set works only over those fields whose characteristic is zero or quasi-polynomially large. To construct the hitting-set, we use the concept of the rank of partial derivative matrix. Unlike previous approaches whose basic building block is a monomial map, we use a polynomial map. The second case we consider is that of commutative ROABP. The best known hitting-set for this case had cost dO(logw)(nw)O(log logw), where d is the individual degree. We improve this hitting-set complexity to (ndw)O(log logw). We get this by achieving rank concentration more efficiently.