{"paper":{"title":"The number of ideals of $\\mathbb{Z}[x]$ containing $x(x-\\alpha)(x-\\beta)$ with given index","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Mitsugu Hirasaka, Semin Oh","submitted_at":"2016-08-30T15:36:43Z","abstract_excerpt":"It is well-known that a connected regular graph is strongly-regular if and only if its adjacency matrix has exactly three eigenvalues. Let $B$ denote an integral square matrix and $\\langle B \\rangle$ denote the subring of the full matrix ring generated by $B$. Then $\\langle B \\rangle$ is a free $\\mathbb{Z}$-module of finite rank, which guarantees that there are only finitely many ideals of $\\langle B \\rangle$ with given finite index. Thus, the formal Dirichlet series $\\zeta_{\\langle B \\rangle}(s)=\\sum_{n\\geq 1}a_n n^{-s}$ is well-defined where $a_n$ is the number of ideals of $\\langle B \\rangl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.08508","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"}