{"paper":{"title":"Quadratic Equations in Hyperbolic Groups are NP-complete","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Alex Taam, Alina Vdovina, Atefeh Mohajeri, Olga Kharlampovich","submitted_at":"2013-06-04T23:14:29Z","abstract_excerpt":"We prove that in a torsion-free hyperbolic group $\\Gamma$, the length of the value of each variable in a minimal solution of a quadratic equation $Q=1$ is bounded by $N|Q|^3$ for an orientable equation, and by $N|Q|^{4}$ for a non-orientable equation, where $|Q|$ is the length of the equation, and the constant $N$ can be computed. We show that the problem, whether a quadratic equation in $\\Gamma$ has a solution, is in NP, and that there is a PSpace algorithm for solving arbitrary equations in $\\Gamma$. If additionally $\\Gamma$ is non-cyclic, then this problem (of deciding existence of a soluti"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.0941","kind":"arxiv","version":4},"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"}