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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 "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2605.25265","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-05-24T21:34:32Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"9d98c86d1f4ac623b34e2d6fa8f45c58d6471672e1cea4b86f2d09050ba5ecbc","abstract_canon_sha256":"494383948220ec3020d94cb8de1ad89a86ead594f8e89e42ca00a42e5d51218e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-26T02:04:26.206929Z","signature_b64":"wSrGW/p9XGpIW3ATsN6C5Jw+xJf+JL0Fp0T632teitavwmwTNbNBPx32DQj33RTT9Ah3JpSpdPNDzQokXuKYAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"40f8421664ade8a860fc3f77bec8849cbd35ce1200286655a1792b7c0c2c48f2","last_reissued_at":"2026-05-26T02:04:26.206078Z","signature_status":"signed_v1","first_computed_at":"2026-05-26T02:04:26.206078Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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