An Algebraic Rigidity Framework for Order-Oblivious Deterministic Black-Box PIT of ROABPs
Shalender Singh, Vishnupriya Singh

TL;DR
This paper introduces an algebraic rigidity framework for ROABPs that enables the first order-oblivious deterministic black-box polynomial identity testing algorithm, significantly advancing derandomization in algebraic complexity.
Contribution
It develops a novel algebraic rigidity approach based on matrix word algebras, enabling order-oblivious deterministic PIT for ROABPs without prior variable order knowledge.
Findings
First order-oblivious deterministic black-box PIT algorithm for ROABPs.
Rigidity reduces algebraic degrees of freedom to dimension at most w^2.
Formulation of a Modular Stability Conjecture to achieve polynomial-time PIT.
Abstract
Deterministic black-box polynomial identity testing (PIT) for read-once oblivious algebraic branching programs (ROABPs) is a central open problem in algebraic complexity, particularly in the absence of variable ordering. Prior deterministic algorithms either rely on order information or incur significant overhead through combinatorial isolation techniques. In this paper, we introduce an algebraic rigidity framework for ROABPs based on the internal structure of their associated matrix word algebras. We show that nonzero width- ROABPs induce word algebras whose effective algebraic degrees of freedom collapse to dimension at most , independent of the number of variables. This rigidity enables deterministic witness construction via intrinsic algebraic invariants, bypassing rank concentration, isolation lemmas, and probabilistic tools used in previous work.Thus, we obtain the first…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsComplexity and Algorithms in Graphs · Logic, programming, and type systems · Advanced Graph Theory Research
