A Complete Classification of Ideal Chomp Games on Low-Rank Algebras
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917056787513344 |
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| author | Karl, Leopold |
| author_facet | Karl, Leopold |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_01673 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Complete Classification of Ideal Chomp Games on Low-Rank Algebras Karl, Leopold Combinatorics Rings and Algebras 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. |
| title | A Complete Classification of Ideal Chomp Games on Low-Rank Algebras |
| topic | Combinatorics Rings and Algebras |
| url | https://arxiv.org/abs/2511.01673 |