{"paper":{"title":"On the rationality of Poincar\\'e series of Gorenstein algebras via Macaulay's correspondence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.AC","authors_text":"Gianfranco Casnati, Joachim Jelisiejew, Roberto Notari","submitted_at":"2013-07-05T17:47:01Z","abstract_excerpt":"Let $A$ be a local Artinian Gorenstein ring with algebraically closed residue field $A/{\\frak M}=k$ of characteristic 0, and let $P_A(z) := \\sum_{p=0}^{\\infty} ({\\mathrm{ Tor}}_p^A(k,k))z^p $ be its Poincar\\'e series. We prove that $P_A(z)$ is rational if either $\\dim_k({{\\frak M}^2/{\\frak M}^3}) \\leq 4 $ and $ \\dim_k(A) \\leq 16,$ or there exist $m\\leq 4$ and $c$ such that the Hilbert function $H_A(n)$ of $A$ is equal to $ m$ for $n\\in [2,c]$ and equal to 1 for $n > c$. The results are obtained thanks to a decomposition of the apolar ideal $\\mathrm {Ann}(F)$ when $F=G+H$ and $G$ and $H$ belong"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.1676","kind":"arxiv","version":2},"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"}