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

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 $(nw)^{O(\log n)}$, 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 $n^{O(\log w)}$. 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 large enough. 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 $d^{O(\log w)}(nw)^{O(\log \log w)}$, where $d$ is the individual degree. We improve this hitting-set complexity to $(ndw)^{O(\log \log w)}$. We get this by achieving rank concentration more efficiently.