🤖 AI Summary
This work addresses the central open problem of deterministic black-box polynomial identity testing (PIT) for read-once oblivious algebraic branching programs (ROABPs) with unknown variable order. It introduces an “algebraic rigidity” framework that constructs deterministic witnesses by exploiting the low-dimensional freedom of the matrix-word algebra associated with ROABPs, without relying on variable ordering or combinatorial isolation. The proposed method yields the first order-oblivious deterministic black-box PIT algorithm with running time \( n \cdot (wd)^{O(w^2)} \); under the modular stability conjecture, this complexity can be further improved to fully polynomial time. The paper also introduces novel tools—including algebraic rigidity, intrinsic invariants, and cyclic quotient ring hashing—offering fresh perspectives in algebraic complexity theory.
📝 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-$w$ ROABPs induce word algebras whose effective algebraic degrees of freedom collapse to dimension at most $w^2$, 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 order-oblivious deterministic black-box PIT algorithm for ROABPs, running in quasi-polynomial time $n\cdot(wd)^{O(w^2)}$. This establishes that algebraic rigidity alone suffices to derandomize PIT in this model, without assuming ordering information. The framework further isolates a single remaining obstacle to full polynomial-time complexity. We formulate a Modular Stability Conjecture, asserting that width-$w$ ROABPs are stable under hashing into cyclic quotient rings $\mathbb{K}[\lambda]/<\lambda^r-1>$ once the modulus exceeds a polynomial threshold in $w$ and the individual degree. This conjecture arises naturally from the low-dimensional coefficient structure revealed by rigidity and is supported by extensive empirical evidence. Assuming the conjecture, our methods yield a fully polynomial-time deterministic black-box PIT algorithm for ROABPs, matching the complexity of the best-known white-box algorithms and reducing the black-box problem to a concrete algebraic stability question.