{"paper":{"title":"Solutions of polynomial equation over $\\mathbb{F}_p$ and new bounds of additive energy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ilya Vyugin, Sergey Makarychev","submitted_at":"2015-03-26T15:34:29Z","abstract_excerpt":"We present a new proof of Corvaja and Zannier's \\cite{C-Z} the upper bound of the number of solutions $(x,y)$ of the algebraic equation $P(x,y)=0$ over a field $\\mathbb{F}_p$ ($p$ is a prime), in the case, where $x\\in g_1G$, $y\\in g_2G$, ($g_1G$, $g_2G$ -- are cosets by some subgroup $G$ of a multiplicative group $\\mathbb{F}_p^*$). The estimate of Corvaja and Zannier was improved in average, and some applications of it has been obtained. In particular we present the new bounds of additive and polynomial energy."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.01354","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"}