关于Sutner猜想的一个解析
Resolution to Sutner's Conjecture
摘要：
考虑一个在简单图$G = (V,E)$上玩的游戏，其中每个顶点由一个可点击的光组成。单击任何顶点$v$可切换$v$及其相邻顶点的开/关状态。一个人通过找到一个关闭所有灯的点击序列来赢得游戏。当$G$是一个$5 \times 5$网格时，这个游戏可以从Tiger Electronics作为Lights Out商业获得。Sutner是第一个用数学方法研究这些博弈的人。他发现，当$d(G) = \text{dim}(\text{ker}(A + I))$在字段$GF(2)$，其中$A$是邻接矩阵的$G$，是0所有的初始配置是可解的。在研究$n \times n$网格图时，Sutner证明了$d_{2n+1} = 2d_{n} + δ_{n}, δ_{n} \in \{0,2\}, δ_{2n+1} = δ_{n}$，其中$d_n = d(G)$用于$G$是$n \times n$网格图。我们肯定地解决了这个猜测。我们使用Sutner的结果，给出$d_n$作为环$\mathbb{Z}_2[x]$中两个多项式的GCD。然后，我们应用Hunziker，Machiavelo和Park的身份，这些身份涉及$(2n+1) \times (2n+1)$网格和$n \times n$网格的多项式。最后，我们使用Ore关于两种产品的GCD的一个结果。这些结果使我们能够证明Sutner猜想。然后我们进一步说明$n$ $δ_n$的值是0还是2。
英文摘要：
Consider a game played on a simple graph $G = (V,E)$ where each vertex consists of a clickable light. Clicking any vertex $v$ toggles the on/off state of $v$ and its neighbors. One wins the game by finding a sequence of clicks that turns off all the lights. When $G$ is a $5 \times 5$ grid, this game was commercially available from Tiger Electronics as Lights Out. Sutner was one of the first to study these games mathematically. He found that when $d(G) = \text{dim}(\text{ker}(A + I))$ over the field $GF(2)$, where $A$ is the adjacency matrix of $G$, is 0 all initial configurations are solvable. When investigating $n \times n$ grid graphs, Sutner conjectured that $d_{2n+1} = 2d_{n} + δ_{n}, δ_{n} \in \{0,2\}, δ_{2n+1} = δ_{n}$, where $d_n = d(G)$ for $G$ an $n \times n$ grid graph. We resolve this conjecture in the affirmative. We use results from Sutner that give $d_n$ as the GCD of two polynomials in the ring $\mathbb{Z}_2[x]$. We then apply identities from Hunziker, Machiavelo, and Park that relate the polynomials of $(2n+1) \times (2n+1)$ grids and $n \times n$ grids. Finally, we use a result from Ore about the GCD of two products. Together these results allow us to prove Sutner's conjecture. We then go further and show for exactly which values of $n$ $δ_n$ is 0 or 2.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
