{"paper":{"title":"The diophantine exponent of the $\\mathbb{Z}/q\\mathbb{Z}$ points of $S^{d-2}\\subset S^d$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Mostafa W. Hassan, Naser T. Sardari, Rodrigo Smith, Xiaohan Zhu, Yuchen Mao","submitted_at":"2018-11-12T00:21:33Z","abstract_excerpt":"Assume a polynomial-time algorithm for factoring integers, Conjecture~\\ref{conj}, $d\\geq 3,$ and $q$ and $p$ are prime numbers, where $p\\leq q^A$ for some $A>0$. We develop a polynomial-time algorithm in $\\log(q)$ that lifts every $\\mathbb{Z}/q\\mathbb{Z}$ point of $S^{d-2}\\subset S^{d}$ to a $\\mathbb{Z}[1/p]$ point of $S^d$ with the minimum height. We implement our algorithm for $d=3 \\text{ and }4$. Based on our numerical results, we formulate a conjecture which can be checked in polynomial-time and gives the optimal bound on the diophantine exponent of the $\\mathbb{Z}/q\\mathbb{Z}$ points of $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.06831","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"}