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Our goal of this paper is to study the Newton polygon of the $L$-functions associated to a finite character of $\\mathbb{Z}_p$ and a generic polynomial whose convex hull is a fixed triangle $\\Delta$. We denote this polygon by $\\textrm{GNP}(\\Delta)$. We prove a lower bound of $\\textrm{GNP}(\\Delta)$, which we call the improved Hodge polygon $\\textrm{IHP}(\\Delta)$, and we conjecture tha"},"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":"1701.00254","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-01-01T15:55:29Z","cross_cats_sorted":[],"title_canon_sha256":"8d005360ba838ae321f75baa3c1b36608156552610d963200f7ff5e005c38eb0","abstract_canon_sha256":"0fec6d0d39b85a6b4b08cd99ba35f9232d5cb2f0d3069f7d3f92b4a3354b8fc6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:53:16.917997Z","signature_b64":"gidXa4jimXiAdNSt+2gF6jksEFUH78AdxWiqRZCu5yRJC/7uKxSTULMuq86DDCQQQJNEtUJnjn9JsaStzvFOCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6ef31df27c5cc327a4cde3620a0405e47df3bbf47a71444aa1546b73f6298de0","last_reissued_at":"2026-05-18T00:53:16.917413Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:53:16.917413Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Generic Newton polygon for exponential sums in two variables with triangular base","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Rufei Ren","submitted_at":"2017-01-01T15:55:29Z","abstract_excerpt":"Let $p$ be a prime number. 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