{"paper":{"title":"Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.SC","authors_text":"George Labahn, Vincent Neiger, Wei Zhou","submitted_at":"2016-07-14T16:01:59Z","abstract_excerpt":"Given a nonsingular $n \\times n$ matrix of univariate polynomials over a field $\\mathbb{K}$, we give fast and deterministic algorithms to compute its determinant and its Hermite normal form. Our algorithms use $\\widetilde{\\mathcal{O}}(n^\\omega \\lceil s \\rceil)$ operations in $\\mathbb{K}$, where $s$ is bounded from above by both the average of the degrees of the rows and that of the columns of the matrix and $\\omega$ is the exponent of matrix multiplication. The soft-$O$ notation indicates that logarithmic factors in the big-$O$ are omitted while the ceiling function indicates that the cost is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.04176","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"}