{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:3FGK36MJKOIWI76B5W3VPQFPW5","short_pith_number":"pith:3FGK36MJ","schema_version":"1.0","canonical_sha256":"d94cadf9895391647fc1edb757c0afb76d3f8012a832c6e8bd7e1664e4f9d0c1","source":{"kind":"arxiv","id":"2605.12903","version":1},"attestation_state":"computed","paper":{"title":"Componentwise height bounds for polynomial value-set lifting","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Rational components with one geometric point at infinity contribute sharp power-log order B^{[k:Q]/d_X(C)} (log B)^{|S|-1} when S-active.","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Henry Shin","submitted_at":"2026-05-13T02:28:01Z","abstract_excerpt":"Let $f,g \\in k[x]$ be nonconstant polynomials over a number field $k$. We count $S$-integer inputs $a$ for which $f(a)$ has a $k$-rational preimage under $g$, after removing the polynomial graph components $Y=h(X)$ with $f=g\\circ h$. The main theorem gives componentwise height bounds. For a rational component of $f(X)-g(Y)=0$ with one geometric point at infinity and projection degree $d_X(C)$ to the $X$-line, the corresponding contribution has the sharp power-log order $B^{[k:\\mathbb{Q}]/d_X(C)}(\\log B)^{q_{k,S}}$, where $q_{k,S}=\\mathrm{rk}\\,\\mathcal{O}_{k,S}^{\\ast}=|S|-1$, precisely when its"},"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":true,"formal_links_present":false},"canonical_record":{"source":{"id":"2605.12903","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-13T02:28:01Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"a9480d13dce943905f9a7b0f18d5ce84e3ec4845c00316fddf5933f42a9bf6ea","abstract_canon_sha256":"ed76e1982b3003ba17d7721bdca813c4e3916995d931b1c37684c06cfd049b8c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:09:10.677455Z","signature_b64":"af7COPI8daQ9bHsZitmvA2r33fkjbzwmjWc0nr5PEnKPigXu+f2HF+QkBHcaCqIYnb0MnpVr97CuuqKuElFIDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d94cadf9895391647fc1edb757c0afb76d3f8012a832c6e8bd7e1664e4f9d0c1","last_reissued_at":"2026-05-18T03:09:10.676671Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:09:10.676671Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Componentwise height bounds for polynomial value-set lifting","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Rational components with one geometric point at infinity contribute sharp power-log order B^{[k:Q]/d_X(C)} (log B)^{|S|-1} when S-active.","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Henry Shin","submitted_at":"2026-05-13T02:28:01Z","abstract_excerpt":"Let $f,g \\in k[x]$ be nonconstant polynomials over a number field $k$. We count $S$-integer inputs $a$ for which $f(a)$ has a $k$-rational preimage under $g$, after removing the polynomial graph components $Y=h(X)$ with $f=g\\circ h$. The main theorem gives componentwise height bounds. For a rational component of $f(X)-g(Y)=0$ with one geometric point at infinity and projection degree $d_X(C)$ to the $X$-line, the corresponding contribution has the sharp power-log order $B^{[k:\\mathbb{Q}]/d_X(C)}(\\log B)^{q_{k,S}}$, where $q_{k,S}=\\mathrm{rk}\\,\\mathcal{O}_{k,S}^{\\ast}=|S|-1$, precisely when its"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For a rational component of f(X)-g(Y)=0 with one geometric point at infinity and projection degree d_X(C) to the X-line, the corresponding contribution has the sharp power-log order B^{[k:Q]/d_X(C)}(log B)^{q_{k,S}}, where q_{k,S}=rk O_{k,S}^* = |S|-1, precisely when its X-parametrization is S-active.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The analysis assumes that after removing the graph components Y=h(X) with f=g o h, the remaining rational components of the curve f(X)-g(Y)=0 can be classified by their number of geometric points at infinity and that the S-activity of the X-parametrization is well-defined and detectable.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Rational components of f(X)-g(Y)=0 with one geometric point at infinity and projection degree d contribute asymptotically B^{[k:Q]/d} (log B)^{|S|-1} when S-active, while others contribute at most polylogarithmically or finitely many terms.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Rational components with one geometric point at infinity contribute sharp power-log order B^{[k:Q]/d_X(C)} (log B)^{|S|-1} when S-active.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"305f4bef83c95b7665a3d3392b9f2911ee0f7a8a819b66c3e187095967936f01"},"source":{"id":"2605.12903","kind":"arxiv","version":1},"verdict":{"id":"44377ef2-a7eb-40ea-ae6e-cdcfc2029789","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:52:06.488609Z","strongest_claim":"For a rational component of f(X)-g(Y)=0 with one geometric point at infinity and projection degree d_X(C) to the X-line, the corresponding contribution has the sharp power-log order B^{[k:Q]/d_X(C)}(log B)^{q_{k,S}}, where q_{k,S}=rk O_{k,S}^* = |S|-1, precisely when its X-parametrization is S-active.","one_line_summary":"Rational components of f(X)-g(Y)=0 with one geometric point at infinity and projection degree d contribute asymptotically B^{[k:Q]/d} (log B)^{|S|-1} when S-active, while others contribute at most polylogarithmically or finitely many terms.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The analysis assumes that after removing the graph components Y=h(X) with f=g o h, the remaining rational components of the curve f(X)-g(Y)=0 can be classified by their number of geometric points at infinity and that the S-activity of the X-parametrization is well-defined and detectable.","pith_extraction_headline":"Rational components with one geometric point at infinity contribute sharp power-log order B^{[k:Q]/d_X(C)} (log B)^{|S|-1} when S-active."},"references":{"count":17,"sample":[{"doi":"10.1142/s1793042109002274","year":2009,"title":"P. 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