  The dimension of partial derivatives (Nisan and Wigderson, 1997) is a popular
measure for proving lower bounds in algebraic complexity. It is used to give
strong lower bounds on the Waring decomposition of polynomials (called Waring
rank). This naturally leads to an interesting open question: does this measure
essentially characterize the Waring rank of any polynomial?
  The well-studied model of Read-once Oblivious ABPs (ROABPs for short) lends
itself to an interesting hierarchy of 'sub-models': Any-Order-ROABPs (ARO),
Commutative ROABPs, and Diagonal ROABPs. It follows from previous works that
for any polynomial, a bound on its Waring rank implies an analogous bound on
its Diagonal ROABP complexity (called the duality trick), and a bound on its
dimension of partial derivatives implies an analogous bound on its 'ARO
complexity': ROABP complexity in any order (Nisan, 1991). Our work strengthens
the latter connection by showing that a bound on the dimension of partial
derivatives in fact implies a bound on the commutative ROABP complexity. Thus,
we improve our understanding of partial derivatives and move a step closer
towards answering the above question.
  Our proof builds on the work of Ramya and Tengse (2022) to show that the
commutative-ROABP-width of any homogeneous polynomial is at most the dimension
of its partial derivatives. The technique itself is a generalization of the
proof of the duality trick due to Saxena (2008).
