{"paper":{"title":"Recovering the cluster picture of a polynomial over a discretely valued field","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Lilybelle Cowland Kellock","submitted_at":"2024-10-22T16:27:31Z","abstract_excerpt":"For $f(x)$ a separable polynomial of degree $d$ over a discretely valued field $K$, we describe how the cluster picture of $f(x)$ over $K$, in other words the set of tuples $\\{(\\mathrm{ord}(x_i-x_j),i,j) : 1\\leq i< j \\leq d \\}$ where $x_1,\\dots,x_d$ are the roots of $f(x)$, can be recovered without knowing the roots of $f(x)$ over $\\bar{K}$. We construct an explicit list of polynomials $g_d^{(1)},\\dots,g_d^{(t_d)}\\in\\mathbb{Z}[A_0,\\dots,A_{d-1}]$ such that the valuations $\\mathrm{ord}(g_{d}^{(i)}(a_0,\\dots,a_{d-1}))$ for $i=1,\\dots,t_d$ uniquely determine this set of distances for the polynomi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.17148","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2410.17148/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"}