{"paper":{"title":"Bounds on the number of connected components for tropical prevarieties","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Alex Davydow, Dima Grigoriev","submitted_at":"2015-11-20T14:25:29Z","abstract_excerpt":"For a tropical prevariety in ${R}^n$ given by a system of $k$ tropical polynomials in $n$ variables with degrees at most $d$, we prove that its number of connected components is less than ${k+7n-1 \\choose 3n} \\cdot \\frac{d^{3n}}{k+n+1}$. On a number of $0$-dimensional connected components a better bound ${k+4n \\choose 3n} \\cdot \\frac{d^n}{k+n+1}$ is obtained, which extends the Bezout bound due to B.~Sturmfels from the the case $k=n$ to an arbitrary $k\\ge n$. Also we show that the latter bound is close to sharp, in particular, the number of connected components can depend on $k$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.06609","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"}