{"paper":{"title":"Zeros of random tropical polynomials, random polytopes and stick-breaking","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.CO","math.NT"],"primary_cat":"math.PR","authors_text":"Francois Baccelli, Ngoc Mai Tran","submitted_at":"2014-03-30T23:14:30Z","abstract_excerpt":"For $i = 0, 1, \\ldots, n$, let $C_i$ be independent and identically distributed random variables with distribution $F$ with support $(0,\\infty)$. The number of zeros of the random tropical polynomials $\\mathcal{T}f_n(x) = \\min_{i=1,\\ldots,n}(C_i + ix)$ is also the number of faces of the lower convex hull of the $n+1$ random points $(i,C_i)$ in $\\mathbb{R}^2$. We show that this number, $Z_n$, satisfies a central limit theorem when $F$ has polynomial decay near $0$. Specifically, if $F$ near $0$ behaves like a $gamma(a,1)$ distribution for some $a > 0$, then $Z_n$ has the same asymptotics as the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.7829","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"}