{"paper":{"title":"Ranks of Quotients, Remainders and $p$-Adic Digits of Matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.SC"],"primary_cat":"math.NT","authors_text":"Andy Novocin, Mark Giesbrecht, Mustafa Elsheikh","submitted_at":"2014-01-26T16:50:00Z","abstract_excerpt":"For a prime $p$ and a matrix $A \\in \\mathbb{Z}^{n \\times n}$, write $A$ as $A = p (A \\,\\mathrm{quo}\\, p) + (A \\,\\mathrm{rem}\\, p)$ where the remainder and quotient operations are applied element-wise. Write the $p$-adic expansion of $A$ as $A = A^{[0]} + p A^{[1]} + p^2 A^{[2]} + \\cdots$ where each $A^{[i]} \\in \\mathbb{Z}^{n \\times n}$ has entries between $[0, p-1]$. Upper bounds are proven for the $\\mathbb{Z}$-ranks of $A \\,\\mathrm{rem}\\, p$, and $A \\,\\mathrm{quo}\\, p$. Also, upper bounds are proven for the $\\mathbb{Z}/p\\mathbb{Z}$-rank of $A^{[i]}$ for all $i \\ge 0$ when $p = 2$, and a conje"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.6667","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"}