{"paper":{"title":"Stable complete coordinates for multisets of points via basic $r$-symmetric tropical polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.CO","math.MG"],"primary_cat":"cs.DM","authors_text":"Susumu Kubo","submitted_at":"2026-06-29T11:58:08Z","abstract_excerpt":"A multiset of $n$ unordered points in $\\mathbb{R}^r$ -- a point cloud, or, for $r=2$, a persistence barcode of birth-death pairs -- is a point of the orbit space $\\mathbb{R}^{nr}/S_n$ for the symmetric group $S_n$ permuting the rows of an $n \\times r$ matrix; a separating family of invariants on this space is exactly a complete set of permutation-independent coordinates. We provide one that is explicit, small, and stable, in the max-plus (tropical) setting: for all $n \\geq 1$ and $r \\geq 1$, the $\\binom{n+r}{r}$ basic $r$-symmetric tropical polynomials, of degree at most $n$, separate the orbi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.30184","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.30184/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}