{"paper":{"title":"Higher dimensional Frobenius problem: Maximal saturated cone, growth function and rigidity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ai-Hua Fan, Hui Rao, Yuan Zhang","submitted_at":"2014-11-26T06:34:50Z","abstract_excerpt":"We consider $m$ integral vectors $X_1,...,X_m \\in \\mathbb{Z}^s$ located in a half-space of $\\mathbb{R}^s$ ($m\\ge s\\geq 1$) and study the structure of the additive semi-group $X_1 \\mathbb{N} +... + X_m \\mathbb{N}$. We introduce and study maximal saturated cone and directional growth function which describe some aspects of the structure of the semi-group. When the vectors $X_1, ..., X_m$ are located in a fixed hyperplane, we obtain an explicit formula for the directional growth function and we show that this function completely characterizes the defining data $(X_1, ..., X_m)$ of the semi-group."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.7118","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"}