{"paper":{"title":"On quantitative aspects of trace polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.GR","authors_text":"Ilya Kapovich","submitted_at":"2026-05-24T21:34:32Z","abstract_excerpt":"By the classic results of Fricke and Klein , for every word $w$ in the free group $F(a,b)$ there exists a unique integer \\emph{trace polynomial} $f_w(x,y,z)\\in Z[x,y,z]$ such that $Tr(w(A,B))=f_w(Tr A,Tr B,Tr AB)$ for all $A,B\\in SL(2,C)$. In this paper we study quantitative aspects of trace polynomials. We prove that for any nontrivial cyclically reduced word $w\\in F(a,b)$ of length $n$ one has $\\frac{n}{2}\\le deg f_w\\le n$, and both bounds are sharp. More precisely, if $k=||w||_{syl}$ is the cyclic syllable length, then $deg f_w\\ge n-k/2$; for positive words $w=a^{\\alpha_1}b^{\\beta_1}\\cdots "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.25265","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.25265/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}