{"paper":{"title":"Constructions for the Elekes-Szab\\'o and Elekes-R\\'onyai problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Audie Warren, Frank de Zeeuw, Mehdi Makhul, Oliver Roche-Newton","submitted_at":"2018-12-03T10:40:54Z","abstract_excerpt":"We give a construction of a non-degenerate polynomial $F\\in \\mathbb R[x,y,z]$ and a set $A$ of cardinality $n$ such that $\\left|Z(F)\\cap (A \\times A \\times A) \\right| \\gg n^{\\frac{3}{2}}$, thus providing a new lower bound construction for the Elekes--Szab\\'o problem. We also give a related construction for the Elekes--R\\'onyai problem restricted to a subgraph. This consists of a polynomial $f\\in \\mathbb R[x,y]$ that is not additive or multiplicative, a set $A$ of size $n$, and a subset $P\\subset A\\times A$ of size $|P|\\gg n^{3/2}$ on which $f$ takes only $n$ distinct values."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.00654","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"}