方格沙堆群中全一类的阶的2-adic赋值
The 2-Adic Valuation of the Order of the All-Ones Class in the Sandpile Group of a Square
- Institute of Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与信息学研究所）
- Department of Mathematics, Middle East Technical University（中东理工大学数学系）
- Guangdong Technion Israel Institute of Technology (GTIIT)（广东以色列理工学院）
- Technion-Israel Institute of Technology（以色列理工学院）
- Center for Research and Advanced Studies of the National Polytechnic Institute (CINVESTAV)（国立理工学院高级研究中心）
- Laboratory of Combinatorial and Geometric Structures, Moscow Institute of Physics and Technology（莫斯科物理技术学院组合与几何结构实验室）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
研究方格沙堆群中全一类的阶的2-adic赋值，通过循环基和多项式理想方法，给出了精确的公式。
中文摘要 AI 辅助
在连通的$n\times n$方格的所有非汇点各放置一粒沙子，设$L(n)$为该操作在沙堆群中的阶。因此$L(n)$是最小的正整数$q$，使得$q$层均匀沙粒层构成倾倒操作的整数组合。我们证明，对于每个$n\ge1$，\\[ \nu_2(L(n))= \begin{cases} 2,&n=1,\\\\ 1,&n\ge2\text{偶数},\\\\ \nu_2(n+1)+2,&n\ge3\text{奇数}. \end{cases} \\] 对于偶数方格，这由Florescu、Morar、Perkinson、Salter和Xu的多米诺-沙堆结果结合一个简短的奇偶性观察得出。对于奇方格，一个幺模循环基将折叠的余核与两个移位切比雪夫多项式的商等同起来，并将全一类映射到$1$。其阶由该多项式理想的常数部分决定，而不仅仅由行列式决定。两个归一化欧几里得余数在$\mathbb F_2$上化简为连续的斐波那契多项式，从而给出精确的赋值。
英文摘要
Place one grain at every nonsink vertex of the wired $n\times n$ square, and let $L(n)$ be the order of this operation in the sandpile group. Thus $L(n)$ is the least positive $q$ for which $q$ uniform grain layers form an integral combination of toppling moves. We prove that, for every $n\ge1$, \[ ν_2(L(n))= \begin{cases} 2,&n=1,\\ 1,&n\ge2\text{ even},\\ ν_2(n+1)+2,&n\ge3\text{ odd}. \end{cases} \] For even squares, this follows from the domino--sandpile results of Florescu, Morar, Perkinson, Salter, and Xu, completed by a short parity observation. For odd squares, a unimodular cyclic basis identifies the folded cokernel with a quotient by two shifted Chebyshev polynomials and sends the all-ones class to $1$. Its order is determined by the constant part of this polynomial ideal, not just by a determinant. Two normalized Euclidean remainders reduce to consecutive Fibonacci polynomials over $\mathbb F_2$, giving the exact valuation.
