{"paper":{"title":"Parametric Polyhedra with at least $k$ Lattice Points: Their Semigroup Structure and the k-Frobenius Problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT","math.OC"],"primary_cat":"math.CO","authors_text":"Iskander Aliev, Jesus A. De Loera, Quentin Louveaux","submitted_at":"2014-09-18T11:09:32Z","abstract_excerpt":"Given an integral $d \\times n$ matrix $A$, the well-studied affine semigroup $\\mbox{ Sg} (A)=\\{ b : Ax=b, \\ x \\in {\\mathbb Z}^n, x \\geq 0\\}$ can be stratified by the number of lattice points inside the parametric polyhedra $P_A(b)=\\{x: Ax=b, x\\geq0\\}$. Such families of parametric polyhedra appear in many areas of combinatorics, convex geometry, algebra and number theory. The key themes of this paper are: (1) A structure theory that characterizes precisely the subset $\\mbox{ Sg}_{\\geq k}(A)$ of all vectors $b \\in \\mbox{ Sg}(A)$ such that $P_A(b) \\cap {\\mathbb Z}^n $ has at least $k$ solutions. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.5259","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"}