{"paper":{"title":"Fast Computation of Smith Forms of Sparse Matrices Over Local Rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"cs.SC","authors_text":"Andy Novocin, B. David Saunders, Mark Giesbrecht, Mustafa Elsheikh","submitted_at":"2012-01-25T20:32:55Z","abstract_excerpt":"We present algorithms to compute the Smith Normal Form of matrices over two families of local rings.\n  The algorithms use the \\emph{black-box} model which is suitable for sparse and structured matrices. The algorithms depend on a number of tools, such as matrix rank computation over finite fields, for which the best-known time- and memory-efficient algorithms are probabilistic.\n  For an $\\nxn$ matrix $A$ over the ring $\\Fzfe$, where $f^e$ is a power of an irreducible polynomial $f \\in \\Fz$ of degree $d$, our algorithm requires $\\bigO(\\eta de^2n)$ operations in $\\F$, where our black-box is assu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.5365","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"}