AI 中文总结
本研究针对环与域的验证问题，给出了完全避开有限单群分类（CFSG）的确定性O(n²)时间算法，解决了环验证的确定性复杂度问题，同时也得到了域验证的同复杂度确定性算法。
AI 中文摘要
我们考虑如下问题：给定两个n×n表，定义集合S（含n个元素）上的二元运算+和·，判定(S,+,·)是否分别构成环或域。近期Dudek、Fischer、Gokaj、Jin、Künnemann、Mao和Redzic（STOC 2026）取得两项（近）最优结果：(1) 验证环的随机化算法，时间复杂度为O(n²log(1/δ))；(2) 验证域的确定性算法，时间复杂度为O(n²)。他们的算法基于Evra、Gadot、Klein和Komargodski（FOCS 2024）的框架，该框架依赖有限单群分类（CFSG）。本研究给出验证环的确定性O(n²)时间算法，解决了该问题的确定性复杂度问题；作为推论，还得到验证域的确定性O(n²)时间算法。我们的算法是初等的，完全避免了CFSG机制。
英文摘要
We consider the following problems: Given two $n \times n$ tables defining binary operations $+$ and $\cdot$ on a set $S$ of $n$ elements, decide whether $(S,+,\cdot)$ forms a ring or, respectively, a field. Recently, Dudek, Fischer, Gokaj, Jin, Künnemann, Mao, and Redzic (STOC 2026) obtained the following two (near-)optimal results: (1) A randomized $O(n^2\log(1/δ))$-time algorithm for verifying rings. (2) A deterministic $O(n^2)$-time algorithm for verifying fields. Their algorithms build on machinery of Evra, Gadot, Klein, and Komargodski (FOCS 2024), which relies on Classification of Finite Simple Groups (CFSG). In this work, we give a deterministic $O(n^2)$-time algorithm for ring verification, resolving the deterministic complexity of this problem. As a corollary, we also obtain a deterministic $O(n^2)$-time algorithm for field verification. Our algorithms are elementary and avoid CFSG machinery entirely.
