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We settle the instance $s=1$ of a conjecture by Akiyama and Peth\\H{o}, stating that the ratio $v_d^{(s)}/v_d^{(0)}$ is an integer for all $d\\ge 2s.$ Moreover we establish the surprisingly simple formula $v_d^{(1)}/v_d^{(0)} = (P_d(3)-2d-1)/4,$ where $P_d(x)$ are the Legendre polyno"},"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":"1405.1530","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-05-07T08:16:52Z","cross_cats_sorted":[],"title_canon_sha256":"f402dce6545eae4b95971d9e89459ba386c18ffa0e1fef41144a717b3698f2db","abstract_canon_sha256":"d5f457413545cd030cff919d7bb2fb542e95b623f214f58b15f210802f6aa9ac"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:52:24.577970Z","signature_b64":"IDQ/iWiY8MHRj2jqwbK4d+8/9elVmdARLiP4WAVVVdUlPV991Nr1xM3MoAjSjo7bPm0G39mtLIV/oXXHM6dqAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"45919a42a290d3d82810b06b49133c7da75024afbcdaeb1830523a2380777877","last_reissued_at":"2026-05-18T02:52:24.577229Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:52:24.577229Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A number theoretic problem on the distribution of polynomials with bounded roots","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Mario Weitzer, Peter Kirschenhofer","submitted_at":"2014-05-07T08:16:52Z","abstract_excerpt":"Let $\\mathcal{E}_d^{(s)}$ denote the set of coefficient vectors $(a_1,\\dots,a_d)\\in \\mathbb{R}^d$ of contractive polynomials $x^d+a_1x^{d-1}+\\dots+a_d\\in \\mathbb{R}[x]$ that have exactly $s$ pairs of complex conjugate roots and let $v_d^{(s)}=\\lambda_d(\\mathcal{E}_d^{(s)})$ be its ($d$-dimensional) Lebesgue measure. We settle the instance $s=1$ of a conjecture by Akiyama and Peth\\H{o}, stating that the ratio $v_d^{(s)}/v_d^{(0)}$ is an integer for all $d\\ge 2s.$ Moreover we establish the surprisingly simple formula $v_d^{(1)}/v_d^{(0)} = (P_d(3)-2d-1)/4,$ where $P_d(x)$ are the Legendre polyno"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.1530","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":"1405.1530","created_at":"2026-05-18T02:52:24.577343+00:00"},{"alias_kind":"arxiv_version","alias_value":"1405.1530v1","created_at":"2026-05-18T02:52:24.577343+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.1530","created_at":"2026-05-18T02:52:24.577343+00:00"},{"alias_kind":"pith_short_12","alias_value":"IWIZUQVCSDJ5","created_at":"2026-05-18T12:28:33.132498+00:00"},{"alias_kind":"pith_short_16","alias_value":"IWIZUQVCSDJ5QKAQ","created_at":"2026-05-18T12:28:33.132498+00:00"},{"alias_kind":"pith_short_8","alias_value":"IWIZUQVC","created_at":"2026-05-18T12:28:33.132498+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/IWIZUQVCSDJ5QKAQWBVUSEZ4PW","json":"https://pith.science/pith/IWIZUQVCSDJ5QKAQWBVUSEZ4PW.json","graph_json":"https://pith.science/api/pith-number/IWIZUQVCSDJ5QKAQWBVUSEZ4PW/graph.json","events_json":"https://pith.science/api/pith-number/IWIZUQVCSDJ5QKAQWBVUSEZ4PW/events.json","paper":"https://pith.science/paper/IWIZUQVC"},"agent_actions":{"view_html":"https://pith.science/pith/IWIZUQVCSDJ5QKAQWBVUSEZ4PW","download_json":"https://pith.science/pith/IWIZUQVCSDJ5QKAQWBVUSEZ4PW.json","view_paper":"https://pith.science/paper/IWIZUQVC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1405.1530&json=true","fetch_graph":"https://pith.science/api/pith-number/IWIZUQVCSDJ5QKAQWBVUSEZ4PW/graph.json","fetch_events":"https://pith.science/api/pith-number/IWIZUQVCSDJ5QKAQWBVUSEZ4PW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IWIZUQVCSDJ5QKAQWBVUSEZ4PW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IWIZUQVCSDJ5QKAQWBVUSEZ4PW/action/storage_attestation","attest_author":"https://pith.science/pith/IWIZUQVCSDJ5QKAQWBVUSEZ4PW/action/author_attestation","sign_citation":"https://pith.science/pith/IWIZUQVCSDJ5QKAQWBVUSEZ4PW/action/citation_signature","submit_replication":"https://pith.science/pith/IWIZUQVCSDJ5QKAQWBVUSEZ4PW/action/replication_record"}},"created_at":"2026-05-18T02:52:24.577343+00:00","updated_at":"2026-05-18T02:52:24.577343+00:00"}