{"paper":{"title":"Polytropes and Tropical Eigenspaces: Cones of Linearity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ngoc M. Tran","submitted_at":"2012-05-14T20:16:12Z","abstract_excerpt":"The map which takes a square matrix $A$ to its polytrope is piecewise linear. We show that cones of linearity of this map form a polytopal fan partition of $\\{R}^{n \\times n}$, whose face lattice is anti-isomorphic to the lattice of complete set of connected relations. This fan refines the non-fan partition of $\\R^{n \\times n}$ corresponding to cones of linearity of the eigenvector map. Our results answer open questions in a previous work with Sturmfels and lead to a new combinatorial classification of polytropes and tropical eigenspaces."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.3186","kind":"arxiv","version":2},"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"}