{"paper":{"title":"The number of rational points on a family of varieties over finite fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Shaofang Hong, Shuangnian Hu","submitted_at":"2016-03-02T15:49:05Z","abstract_excerpt":"Let $\\mathbb{F}_q$ stand for the finite field of odd characteristic $p$ with $q$ elements ($q=p^{n},n\\in \\mathbb{N} $) and $\\mathbb{F}_q^*$ denote the set of all the nonzero elements of $\\mathbb{F}_{q}$. Let $m$ and $t$ be positive integers. In this paper, by using the Smith normal form of the exponent matrix, we obtain a formula for the number of rational points on the variety defined by the following system of equations over $\\mathbb{F}_{q}$: $$ \\sum\\limits_{j=0}^{t-1}\\sum\\limits_{i=1}^{r_{j+1}-r_j} a_{k,r_j+i}x_1^{e^{(k)}_{r_j+i,1}}...x_{n_{j+1}}^{e^{(k)}_{r_j+i,n_{j+1}}}=b_k, \\ k=1,...,m. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.00760","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"}