用于ROABPs确定性黑盒多项式恒等测试的代数刚性框架
An Algebraic Rigidity Framework for Order-Oblivious Deterministic Black-Box PIT of ROABPs
AI总结：
本文提出了一种基于代数刚性的框架，实现了ROABPs的无序确定性黑盒多项式恒等测试，运行时间接近准多项式时间，并通过模稳定性猜想推动了多项式复杂度的实现。
摘要：
确定性黑盒多项式恒等测试（PIT）对于读一次无偏代数分支程序（ROABPs）是代数复杂性中的核心开放问题，特别是在没有变量顺序信息的情况下。先前的确定性算法要么依赖于顺序信息，要么通过组合隔离技术产生显著的开销。在本文中，我们引入了一个基于其关联矩阵词代数内部结构的ROABPs代数刚性框架。我们证明非零宽度-w的ROABPs诱导的词代数的有效代数自由度会坍缩到至多w²的维度，与变量数量无关。这种刚性使得通过内在的代数不变量进行确定性见证构造成为可能，从而绕过了之前工作中使用的秩集中、隔离引理和概率工具。因此，我们获得了第一个无序的确定性黑盒PIT算法，其运行时间是准多项式时间n·(wd)^{O(w²)}。这表明仅凭代数刚性就足以在该模型中去随机化PIT，而无需假设顺序信息。该框架进一步隔离了实现多项式复杂度的唯一剩余障碍。我们提出了一个模稳定性猜想，断言当模数超过w和个体次数的多项式阈值时，宽度-w的ROABPs在哈希到循环商环K[λ]/<λ^r-1>时是稳定的。这个猜想自然地源于刚性揭示的低维系数结构，并且得到了广泛的经验证据的支持。假设该猜想，我们的方法得出了一种完全多项式时间的确定性黑盒PIT算法，其复杂度与目前最好的白盒算法相同，并将黑盒问题减少到一个具体的代数稳定性问题。
英文摘要：
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 $\mathbb{K}[λ]/< λ^r-1 >$ 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.
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