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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1207.0164","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-06-30T22:56:55Z","cross_cats_sorted":[],"title_canon_sha256":"3036e30afbe492ac2543c613e926de215e669080718f52fa388aac35d08d4808","abstract_canon_sha256":"a9abd438b504d5462138a8cf5c24b022ab421025809c1f6da9f5dd141a6655b2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:12:58.965756Z","signature_b64":"vdpAZhfEDyG5eBuQRbcRfF9v3HnMuhnbk8HTajRfTz/i5nxYFlLmvJjVOJ2KFXLf9muiOT2q7oSWBhzJong/Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c9ecaeacdca2cd81bb4ad66f2fb011ac66fa1fa2482066009db09274ea66e296","last_reissued_at":"2026-05-18T01:12:58.965334Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:12:58.965334Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\K$. 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