{"paper":{"title":"Sumsets and Veronese varieties","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC","math.CO"],"primary_cat":"math.AG","authors_text":"Joan Elias, Liena Colarte-G\\'omez, Rosa M. Mir\\'o-Roig","submitted_at":"2022-02-02T16:20:14Z","abstract_excerpt":"In this paper, to any subset $\\mathcal{A} \\subset \\mathbb{Z}^{n}$ we explicitly associate a unique monomial projection $Y_{n,d_{\\mathcal{A}}}$ of a Veronese variety, whose Hilbert function coincides with the cardinality of the $t$-fold sumsets $t\\mathcal{A}$. This link allows us to tackle the classical problem of determining the polynomial $p_{\\mathcal{A}} \\in \\mathbb{Q}[t]$ such that $|t\\mathcal{A}| = p_{\\mathcal{A}}(t)$ for all $t \\geq t_0$ and the minimum integer $n_0(\\mathcal{A}) \\leq t_0$ for which this condition is satisfied, i.e. the so-called {\\em phase transition} of $|t\\mathcal{A}|$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2202.01114","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2202.01114/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}