{"paper":{"title":"On the volume and the number of lattice of some semialgebraic sets","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.AG","authors_text":"Ha Huy Vui, Tran Gia Loc","submitted_at":"2015-02-21T10:15:48Z","abstract_excerpt":"Let $f = (f_1,\\ldots,f_m) : \\R^n \\longrightarrow \\R^m$ be a polynomial map; $G^f(r) = \\{x\\in\\R^n : |f_i(x)| \\leq r,\\ i =1,\\ldots, m\\}$. We show that if $f$ satisfies the Mikhailov - Gindikin condition then \\begin{itemize} \\item[(i)] $\\text{Volume}\\ G^f(r) \\asymp r^\\theta (\\ln r)^k$ \\item[(ii)] $\\text{Card}\\left(G^f(r) \\cap \\overset{o}{\\ \\Z^n}\\right) \\asymp r^{\\theta'}(\\ln r)^{k'}$, as $r\\to \\infty$, \\end{itemize} where the exponents $\\theta,\\ k,\\ \\theta',\\ k'$ are determined explicitly in terms of the Newton polyhedra of $f$. \\\\ \\indent Moreover, the polynomial maps satisfy the Mikhailov - Gin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.06091","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"}