{"paper":{"title":"Depth of initial ideals of normal edge rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Akihiro Higashitani, Augustine B. O'Keefe, Kyouko Kimura, Takayuki Hibi","submitted_at":"2011-01-21T01:20:43Z","abstract_excerpt":"Let $G$ be a finite graph on the vertex set $[d] = \\{1, ..., d \\}$ with the edges $e_1, ..., e_n$ and $K[\\tb] = K[t_1, ..., t_d]$ the polynomial ring in $d$ variables over a field $K$. The edge ring of $G$ is the semigroup ring $K[G]$ which is generated by those monomials $\\tb^e = t_it_j$ such that $e = \\{i, j\\}$ is an edge of $G$. Let $K[\\xb] = K[x_1, ..., x_n]$ be the polynomial ring in $n$ variables over $K$ and define the surjective homomorphism $\\pi : K[\\xb] \\to K[G]$ by setting $\\pi(x_i) = \\tb^{e_i}$ for $i = 1, ..., n$. The toric ideal $I_G$ of $G$ is the kernel of $\\pi$. It will be pro"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.4058","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}