{"paper":{"title":"A Symbolic Homotopy Algorithm for Solving Composable Polynomial Systems","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AC","math.AG"],"primary_cat":"cs.SC","authors_text":"Thi Xuan Vu","submitted_at":"2026-05-21T14:08:09Z","abstract_excerpt":"We study the problem of computing the isolated regular solutions of a system \\((f_1,\\ldots,f_n)\\) of \\(n\\) polynomial equations in \\(n\\) variables \\((X_1, \\dots, X_n)\\) over a field of characteristic zero \\(k\\). We focus on systems with a \\emph{composable structure}, where each polynomial \\(f_i\\) can be expressed as a composition \\( f_i = h_i(g_1,\\dots,g_n)\\). Exploiting this structure allows us to reduce the original system to one in the \\(g_j\\) variables, thereby significantly improving the efficiency of symbolic solution algorithms. We present a probabilistic algorithm that computes all iso"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.22514","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/2605.22514/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"}