{"paper":{"title":"Points on curves in small boxes en applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ana Zumalac\\'arregui, Igor E. Shparlinski, Javier Cilleruelo, Jos\\'e Hern\\'andez, Mei-Chu Chang, Moubariz Z. Garaev","submitted_at":"2011-11-07T11:13:31Z","abstract_excerpt":"We introduce several new methods to obtain upper bounds on the number of solutions of the congruences $f(x) \\equiv y \\pmod p$ and $f(x) \\equiv y^2 \\pmod p,$ with a prime $p$ and a polynomial $f$, where $(x,y)$ belongs to an arbitrary square with side length $M$. We use these results and methods to derive non-trivial upper bounds for the number of hyperelliptic curves $Y^2=X^{2g+1} + a_{2g-1}X^{2g-1} +...+ a_1X+a_0$ over the finite field $\\F_p$ of $p$ elements, with coefficients in a $2g$-dimensional cube $ (a_0,..., a_{2g-1})\\in [R_0+1,R_0+M]\\times...\\times [R_{2g-1}+1,R_{2g-1}+M]$ that are is"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.1543","kind":"arxiv","version":3},"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"}