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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1401.6667","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-01-26T16:50:00Z","cross_cats_sorted":["cs.SC"],"title_canon_sha256":"bc3ccb6e44fed029f3b4491674729a50dbf87b168d0d69bba198b33ff05ccc04","abstract_canon_sha256":"a05929716738850f462389f3b2bab563259581c11da0a446643f8a8a3c1d596c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:00:36.609638Z","signature_b64":"Hz78SsjHf03A/XLBtZnGKLw9CesjpbxwQaEx33WB4LxQaRh254B1k/OKWGz/oBaNBMP+TrQfZg1ctSnxbUGnDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"99805c8896c7a0fd7e22853948ef037641225d2dfebe625e205d4ace1c8eacc5","last_reissued_at":"2026-05-18T03:00:36.608880Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:00:36.608880Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $A \\,\\mathrm{quo}\\, p$. 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