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Suppose that $M_*(P,d)$ is the smallest mode of $P(x+d)$, and $M^*(P,d)$ the greatest mode. Wang and Yeh conjectured that if $d_2>d_1>0$, then $M_*(P,d_1)\\geq M_*(P,d_2)$ and $M^*(P,d_1)\\geq M^*(P,d_2)$. We give a proof of this conjecture."},"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":"1008.4927","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-08-29T14:34:10Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"eadfeedcfe6fb504ae3a94945ddbc6e684fbacc40e8208c9e248db2a9d6a0beb","abstract_canon_sha256":"67fc74816139f21fadb320244568804e1689dd42eec6ef8c4e742746a0cb6c7d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:41:43.404042Z","signature_b64":"LQ7xrqp9KWpuR24nvDRyMrdgQmonmBkDe4pYaCcprrs6gBFuyF7I8rL1zIhA4IRRSUgR1bmjTaU7sN9mafsmBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c7a1a1a0331aa43ee059dd34b5e7e7b706ca37ccf5e5ea4a37b384983a499131","last_reissued_at":"2026-05-18T04:41:43.403553Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:41:43.403553Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Modes of Polynomials Derived from Nondecreasing Sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.CO","authors_text":"Arthur L. B. Yang, Donna Q. J. Dou","submitted_at":"2010-08-29T14:34:10Z","abstract_excerpt":"Wang and Yeh proved that if $P(x)$ is a polynomial with nonnegative and nondecreasing coefficients, then $P(x+d)$ is unimodal for any $d>0$. A mode of a unimodal polynomial $f(x)=a_0+a_1x+\\cdots + a_mx^m$ is an index $k$ such that $a_k$ is the maximum coefficient. Suppose that $M_*(P,d)$ is the smallest mode of $P(x+d)$, and $M^*(P,d)$ the greatest mode. Wang and Yeh conjectured that if $d_2>d_1>0$, then $M_*(P,d_1)\\geq M_*(P,d_2)$ and $M^*(P,d_1)\\geq M^*(P,d_2)$. We give a proof of this conjecture."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.4927","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1008.4927","created_at":"2026-05-18T04:41:43.403629+00:00"},{"alias_kind":"arxiv_version","alias_value":"1008.4927v1","created_at":"2026-05-18T04:41:43.403629+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.4927","created_at":"2026-05-18T04:41:43.403629+00:00"},{"alias_kind":"pith_short_12","alias_value":"Y6Q2DIBTDKSD","created_at":"2026-05-18T12:26:17.028572+00:00"},{"alias_kind":"pith_short_16","alias_value":"Y6Q2DIBTDKSD5YCZ","created_at":"2026-05-18T12:26:17.028572+00:00"},{"alias_kind":"pith_short_8","alias_value":"Y6Q2DIBT","created_at":"2026-05-18T12:26:17.028572+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/Y6Q2DIBTDKSD5YCZ3U2LLZ7HW4","json":"https://pith.science/pith/Y6Q2DIBTDKSD5YCZ3U2LLZ7HW4.json","graph_json":"https://pith.science/api/pith-number/Y6Q2DIBTDKSD5YCZ3U2LLZ7HW4/graph.json","events_json":"https://pith.science/api/pith-number/Y6Q2DIBTDKSD5YCZ3U2LLZ7HW4/events.json","paper":"https://pith.science/paper/Y6Q2DIBT"},"agent_actions":{"view_html":"https://pith.science/pith/Y6Q2DIBTDKSD5YCZ3U2LLZ7HW4","download_json":"https://pith.science/pith/Y6Q2DIBTDKSD5YCZ3U2LLZ7HW4.json","view_paper":"https://pith.science/paper/Y6Q2DIBT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1008.4927&json=true","fetch_graph":"https://pith.science/api/pith-number/Y6Q2DIBTDKSD5YCZ3U2LLZ7HW4/graph.json","fetch_events":"https://pith.science/api/pith-number/Y6Q2DIBTDKSD5YCZ3U2LLZ7HW4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Y6Q2DIBTDKSD5YCZ3U2LLZ7HW4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Y6Q2DIBTDKSD5YCZ3U2LLZ7HW4/action/storage_attestation","attest_author":"https://pith.science/pith/Y6Q2DIBTDKSD5YCZ3U2LLZ7HW4/action/author_attestation","sign_citation":"https://pith.science/pith/Y6Q2DIBTDKSD5YCZ3U2LLZ7HW4/action/citation_signature","submit_replication":"https://pith.science/pith/Y6Q2DIBTDKSD5YCZ3U2LLZ7HW4/action/replication_record"}},"created_at":"2026-05-18T04:41:43.403629+00:00","updated_at":"2026-05-18T04:41:43.403629+00:00"}