{"paper":{"title":"Incidences between points and generalized spheres over finite fields and related problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Le Anh Vinh, Nguyen Duy Phuong, Pham Van Thang","submitted_at":"2014-10-29T08:32:20Z","abstract_excerpt":"Let $\\mathbb{F}_q$ be a finite field of $q$ elements where $q$ is a large odd prime power and $Q =a_1 x_1^{c_1}+...+a_dx_d^{c_d}\\in \\mathbb{F}_q[x_1,...,x_d]$, where $2\\le c_i\\le N$, $\\gcd(c_i,q)=1$, and $a_i\\in \\mathbb{F}_q$ for all $1\\le i\\le d$. A $Q$-sphere is a set of the form $\\lbrace x\\in \\mathbb{F}_q^d | Q(x-b)=r\\rbrace$, where $b\\in \\mathbb{F}_q^d, r\\in \\mathbb{F}_q$. We prove bounds on the number of incidences between a point set $\\mathcal{P}$ and a $Q$-sphere set $\\mathcal{S}$, denoted by $I(\\mathcal{P},\\mathcal{S})$, as the following.\n  $$| I(\\mathcal{P},\\mathcal{S})-\\frac{|\\mathca"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.7899","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"}