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Explicit Commutative ROABPs from Partial Derivatives

Authors Vishwas Bhargava , Anamay Tengse

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Author Details

Vishwas Bhargava
  • Department of Computing and Mathematical Sciences, Caltech, Pasadena, CA, USA
Anamay Tengse
  • School of Computer Sciences, NISER, Bhubaneswar, India

Funding

  • Bhargava, Vishwas: Part of this work was done as a postdoc at University of Waterloo, Canada.
  • Tengse, Anamay: Part of this work was done as a postdoc at Reichman University, Herzliya.

Acknowledgements

VB thanks Rafael Oliveira and Abhiroop Sanyal for numerous insightful discussions. AT thanks Prerona Chatterjee, C Ramya, and Ramprasad Saptharishi for several fruitful discussions about ROABPs over the recent years. AT is also deeply grateful to Ramprasad Saptharishi, Susmita Biswas and Lulu, for hosting him during a part of this work.

Cite As Get BibTex

Vishwas Bhargava and Anamay Tengse. Explicit Commutative ROABPs from Partial Derivatives. In 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 323, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.FSTTCS.2024.10

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).

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • Partial derivatives
  • Apolar ideals
  • Commuting matrices
  • Branching programs

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