{"paper":{"title":"Applications of degree estimate for subalgebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC","math.AG"],"primary_cat":"math.RA","authors_text":"Jie-Tai Yu, Yun-Chang Li","submitted_at":"2010-11-30T14:13:20Z","abstract_excerpt":"Let $K$ be a field of positive characteristic and $K<x, y>$ be the free algebra of rank two over $K$. Based on the degree estimate done by Y.-C. Li and J.-T. Yu, we extend the results of S.J. Gong and J.T. Yu's results: (1) An element $p(x,y)\\in K<x,y>$ is a test element if and only if $p(x,y)$ does not belong to any proper retract of $K<x,y>$; (2) Every endomorphism preserving the automorphic orbit of a nonconstant element of $K<x,y>$ is an automorphism; (3) If there exists some injective endomorphism $\\phi$ of $K<x,y>$ such that $\\phi(p(x,y))=x$ where $p(x,y)\\in K<x,y>$, then $p(x,y)$ is a c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.6551","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"}