{"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"}