{"paper":{"title":"Lattice-point generating functions for free sums of convex sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Matthias Beck, Pallavi Jayawant, Tyrrell B. McAllister","submitted_at":"2012-06-30T22:56:55Z","abstract_excerpt":"Let $\\J$ and $\\K$ be convex sets in $\\R^{n}$ whose affine spans intersect at a single rational point in $\\J \\cap \\K$, and let $\\J \\oplus \\K = \\conv(\\J \\cup \\K)$. We give formulas for the generating function {equation*} \\sigma_{\\cone(\\J \\oplus \\K)}(z_1,..., z_n, z_{n+1}) = \\sum_{(m_1,..., m_n) \\in t(\\J \\oplus \\K) \\cap \\Z^{n}} z_1^{m_1}... z_n^{m_n} z_{n+1}^{t} {equation*} of lattice points in all integer dilates of $\\J \\oplus \\K$ in terms of $\\sigma_{\\cone \\J}$ and $\\sigma_{\\cone \\K}$, under various conditions on $\\J$ and $\\K$. This work is motivated by (and recovers) a product formula of B.\\ B"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.0164","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"}