Explicit Commutative ROABPs from Partial Derivatives

TL;DR

This paper establishes a new connection between the dimension of partial derivatives and the complexity of commutative ROABPs, advancing understanding of algebraic complexity measures and their relationships.

Contribution

It proves that bounds on the dimension of partial derivatives imply bounds on commutative ROABP complexity, strengthening previous connections in algebraic complexity theory.

Findings

Bound on partial derivatives dimension implies bound on commutative ROABP complexity.

Improves understanding of algebraic complexity measures.

Generalizes the duality trick for homogeneous polynomials.

Abstract

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).