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).

翻译：偏导数维度（Nisan与Wigderson，1997）是代数复杂性理论中证明下界的常用度量。该度量被用于给出多项式华林分解（称为华林秩）的强下界，这自然引出一个重要的开放问题：该度量是否本质上刻画了任意多项式的华林秩？经过深入研究的"单次 oblivious 代数分支程序"（简称ROABP）模型衍生出一系列有趣的"子模型"层级结构：任意顺序ROABP（ARO）、交换ROABP与对角ROABP。已有研究表明：对任意多项式，其华林秩的上界可推出对角ROABP复杂度的对应上界（称为对偶技巧）；其偏导数维度的上界则可推出任意顺序ROABP复杂度（Nisan，1991）的对应上界。本研究通过证明偏导数维度的上界实际上可推出交换ROABP复杂度的上界，从而强化了后者的关联性。这不仅深化了我们对偏导数的理解，更为解答上述问题推进了关键一步。我们的证明基于Ramya与Tengse（2022）的工作，通过推广Saxena（2008）提出的对偶技巧证明方法，最终证得任意齐次多项式的交换ROABP宽度至多不超过其偏导数维度。