低秩代数上理想Chomp博弈的完整分类
A Complete Classification of Ideal Chomp Games on Low-Rank Algebras
AI总结：
本文完整分类了秩至多为6的$\ar{K}$-代数上Ideal Chomp Game的获胜策略，证明除五种特定情况外玩家A均有获胜策略，方法结合了博弈论分析、Artinian环结构理论与计算验证。
摘要：
我们完整分类了在秩至多为6的$\ar{K}$-代数R上进行Ideal Chomp Game的获胜策略。在这个双人组合博弈中，玩家交替添加生成元以在给定环R内构建理想，构建出等于整个环的理想者输。我们证明，除了五种特定情况外，玩家A在所有秩至多为6的$\ar{K}$-代数R上都有获胜策略：$\ar{K}$本身、$\ar{K}[x, y]/(x, y)^2$以及另外三个局部代数。我们的方法结合了博弈论分析、Artinian环的结构理论和计算验证。我们还讨论了Henson关于Ideal Chomp Game获胜策略的经典结果，以及关于高维$\ar{K}$-代数上Ideal Chomp Game的思路和开放问题。
英文摘要：
We completely classify winning strategies in the Ideal Chomp Game played on $\bar{K}$-algebras R of rank at most 6. In this two-player combinatorial game, players alternately add generators to build an ideal inside a given ring R, with the player who builds an ideal equal to the entire ring losing. We prove that player A has a winning strategy on all $\bar{K}$-algebras R up to rank 6 except for five specific cases: $\bar{K}$ itself, $\bar{K}[x, y]/(x, y)^2$, and three other local algebras. Our methods combine game-theoretic analysis with the structure theory of Artinian rings and computational verification. We also discuss a classical result of Henson on winning strategies in the Ideal Chomp Game, as well as ideas and open questions about the Ideal Chomp Game on higher-dimensional $\bar{K}$-algebras.
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。
