{"paper":{"title":"The Hilbert series and $a$-invariant of circle invariants","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC","math.CO"],"primary_cat":"math.RA","authors_text":"Christopher Seaton, Daniel Herden, Hans-Christian Herbig, L.Emily Cowie","submitted_at":"2017-07-11T05:00:40Z","abstract_excerpt":"Let $V$ be a finite-dimensional representation of the complex circle $\\mathbb{C}^\\times$ determined by a weight vector $\\mathbf{a}\\in\\mathbb{Z}^n$. We study the Hilbert series $\\operatorname{Hilb}_{\\mathbf{a}}(t)$ of the graded algebra $\\mathbb{C}[V]^{\\mathbb{C}_{\\mathbf{a}}^\\times}$ of polynomial $\\mathbb{C}^\\times$-invariants in terms of the weight vector $\\mathbf{a}$ of the $\\mathbb{C}^\\times$-action. In particular, we give explicit formulas for $\\operatorname{Hilb}_{\\mathbf{a}}(t)$ as well as the first four coefficients of the Laurent expansion of $\\operatorname{Hilb}_{\\mathbf{a}}(t)$ at $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.03128","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"}