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We show that there exists a positive integer $n_{d}$, depending only on $d$, such that if $h(t)$ is a polynomial of degree at most $d$ with nonnegative integer coefficients and $h(0) \\geq 1$, then for $n \\geq n_{d}$, $\\U_{n}h(t)$ has simple, real,"},"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":"0804.3639","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2008-04-23T04:20:32Z","cross_cats_sorted":[],"title_canon_sha256":"c90b4461e74355d34a3bc89cc6f5fea8e9f92f0854d144862c709ad1c8ef7e90","abstract_canon_sha256":"4dc8e797934d18ee7e87752532953996f291f3dc5ef96f883da684dcff734cb7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:41:36.884935Z","signature_b64":"7OYtgd6CPiBle6u242sYSBuzKKikM+L0nnMRwcpVIuEvYNpf8zOXfVMpyOLlNL8NBDyQU5XZms3H/2vHW6oXBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f989ea918f05a2984dcbd51215c1d4f59f0d8c623bfb09503aa3e0687d0738ef","last_reissued_at":"2026-05-18T04:41:36.884475Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:41:36.884475Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Log-Concavity of Hilbert Series of Veronese Subrings and Ehrhart Series","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alan Stapledon, Matthias Beck","submitted_at":"2008-04-23T04:20:32Z","abstract_excerpt":"For every positive integer $n$, consider the linear operator $\\U_{n}$ on polynomials of degree at most $d$ with integer coefficients defined as follows: if we write $\\frac{h(t)}{(1 - t)^{d + 1}} = \\sum_{m \\geq 0} g(m) t^{m}$, for some polynomial $g(m)$ with rational coefficients, then $\\frac{\\U_{n}h(t)}{(1- t)^{d + 1}} = \\sum_{m \\geq 0} g(nm) t^{m}$. 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