{"paper":{"title":"Deterministic Polynomial-time Exact-root Computation for Sparse Polynomials with Bounded Total Degree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"cs.DS","authors_text":"Qiao-Long Huang, Ruichen Qiu, Xiao-Shan Gao, Yichuan Cao","submitted_at":"2026-07-02T16:06:29Z","abstract_excerpt":"We study the problem of deterministically computing the exact root of a sparse polynomial in the multivariate setting. Let $f \\in \\F[x_1,\\ldots,x_n]$ be a nonzero polynomial that is an exact $e$-th power, say $f = g^e$. Suppose $f$ is $s$-sparse, has an individual degree of at most $d$, and a total degree of $D = \\tdeg(f)$. We prove a sparsity bound on the base polynomial $g$: \\[ \\|g\\|_0 \\le s^{D(2d+2)/e + 1}. \\] Based on this bound, we develop a deterministic algorithm that computes the base $g$. %\nIn contrast to the general deterministic factorization algorithm of Bhargava, Saraf, and Volkov"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.02364","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/2607.02364/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"}