[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85014-en":3,"doc-seo-85014-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},85014,13056703019404,"Miles","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Small Matrices with Large Inverses: Unimodular 4×4 Cases","The work studies how close a unimodular n×n integer matrix can approach singularity by tracking the maximum absolute entry α=‖M‖ and the maximum absolute entry of the inverse β=‖M−1‖. For n=4, progress is made by fixing (2k+1)-ary and (k+1)-ary families, and by analyzing 3×3 submatrices A obtained by deleting one row and one column. Determinant bounds for |detA| in nonnegative and signed cases are proved and shown sharp.","arXiv :2607 .07688v1 [math .CO] 8 Jul 2026  \nSmall Matrices with Large Inverses: Unimodular 4 × 4 Cases  \nSteven Finch  \nJuly 8, 2026  \nAbstract. How close to singularity can an n × n unimodular matrix be? For ternary cases as n increases, exact expressions are unlikely, but upon fixing n = 4 and assessing (2k+1)-ary cases as k increases, we make significant progress; similarly for (k+1)-ary cases of 4×4 nonnegative unimodular matrices.  \nAn integer matrix M with determinant ±1 is called unimodular. Let α = ∥M∥ and β = ∥M−1∥, the maximum absolute entry of M and M −1, respectively. If n = 2, then α = β trivially. If n ≥ 2, the ternary lower-triangular n × n matrix  \n􀀰 1 0 0 ... 0 0 􀀱  \n􀁂 −1 1 0 0 0 􀁃  \n􀁂 −1 −1 1 0 0 􀁃  \n􀁂 􀁃  \n􀁂 .. . . .. 􀁃  \n􀁂 . . . 􀁃  \n􀁂 −1 −1 −1 1 0 􀁃􀁀 −1 −1 −1 ... −1 1 􀁁  \nhas α = 1, β = 2n−2 . Define  \nγ(α) = max 􀀈β ≥ 1 : ∃ unimodular M ∈ Zn×n with ∥M∥ = α , 􀀍 M −1 􀀍 = β 􀀉 .  \nWhile γ(k) = k for k ≥ 1 and γ(1) = 2n−2 for 3 ≤ n ≤ 4, other values γ(k) are less clear. The sequence γ(1) = 10, γ(1) = 30, γ(1) = 130, ... defies description (we know only that 442 ≤ γ(1) ≤ 576 [1, 2]) . Define also  \nγ+n(α) = max 􀀈β ≥ 1 : ∃ unimodular M ∈ Z×0n with ∥M∥ = α , 􀀍 M −1 􀀍 = β 􀀉 ;  \nthe alphabet underlying M here has k +1 symbols rather than 2k +1 . The sequence γ+3(1) = 1, γ(1) = 2, γ+5(1) = 3, γ+6(1) = 5, γ+7(1) = 9, γ+8(1) = 18, γ+9(1) = 42, ... again defies description (we know 110 ≤ γ+10(1) ≤ 144 [3]) .  \nOur interest is in the case n = 4 . Let  β := γ(k), β+k := γ(k)  \n0 Copyright © 2026 by Steven R. Finch. All rights reserved.  \nSmall Matrices with Large Inverses: Unimodular 4 × 4 Cases 2  \nand A be a 3 × 3 submatrix of M (delete one row and one column) . For A ∈ Z3×3 let g (A) be the greatest common divisor (gcd) of its nine 2 × 2 minors.  \nTheorem. Let k ≥ 2.  \n(i) Nonnegative case. If all entries of M lie in {0 , 1 ,..., k}, then |detA| ≤ β+k = k(2k2 − 2k + 1) = 2k3 − 2k2 + k.  \n(ii) Signed case. If all entries of M lie in {−k,..., −1 , 0 , 1 ,..., k}, then |detA| ≤ β = k(2k − 1)(2k + 1) = 4k3 − k.  \nBoth bounds are attained for every k ≥ 2.  \nScenario (i) is not a special case of (ii): a nonnegative matrix obeys both bounds, but the sharp bound for the nonnegative subproblem is the smaller β+k, with its own extremal family. The two are genuinely distinct extremal problems sharing the scaffold described below.  \nWe write the rows of A as r 1 , r2 , r3 , and for a row index i let w = w (i) be the cofactor vector (the 2 × 2 minors involving the other two rows, with signs), sodetA = ri · w. We assume WLOG that detA > 0 throughout, since swapping two rows of A negates the determinant while leaving ∥A∥ and every |2×2 minor| – hence g (A)– unchanged.  \n0. The common core  \n0.1. Multilinear cube bound. The determinant is linear in each entry, so over the box its maximum is at a vertex [4], where it equals k 3 multiplied by the largest 3 × 3 determinant with 0/1 (respectively 0/ ± 1) entries [2, 3]:  \ndetA ≤ 2k3 (nonnegative) , detA ≤ 4k3 (signed) .  \nCall these Hadamard-type ceilings. Likewise maxri ri · w = k∥w∥1 is itself such a determinant, giving the cofactor bounds  \n∥w∥1 ≤ 2k2 (nonnegative) , ∥w∥1 ≤ 4k2 (signed) , |wj | ≤ 2k2 .  \n0.2. Embedding. This property is used identically in both §1 and §2.  \nLemma E. If M ∈ Zn×n is unimodular and A is any (n−1) × (n−1) submatrix, then g(A) = 1 .  \nProof. Deleting one row and one column is the same as conjugating by permutation matrices (preserving unimodularity, A, and g (A)) and deleting the last row/column. Hence assume  \nM = 􀀒 ud􀀓 , det M = ±1 .  \nSmall Matrices with Large Inverses: Unimodular 4 × 4 Cases 3  \nThe one-row/one-column bordering identity (a polynomial identity, verified where A is invertible via A −1 = adj(A)/detA) is  \ndetM = ddet(A) − v⊤ adj(A)u. (⋆)  \nThe entries of adj(A) are ± the 2×2 minors, thus g(A) divides them all; and detA = Ph a 1h adj(A)h1 shows g(A) | detA. Therefore g(A) | det M = ±1, i.e. , g (A) = 1 . ■  \nFixing the border data (d, v , u)","cbCaipRppBaT91UP","https://ap.wps.com/l/cbCaipRppBaT91UP","pdf",298796,1,20,"English","en",105,"# Abstract\n# Problem Setup and Definitions\n## Parameters α, β, γ\n# n = 4 Framework\n## Submatrix A and gcd of minors g(A)\n# Main Theorem Bounds\n## Nonnegative case\n## Signed case\n# Shared Architecture\n## Multilinear cube bound\n## Embedding and Lemma E\n# Nonnegative Case (A ∈ {0,…,k}³×³)\n## Cross product lemma","[{\"question\":\"What does the parameter γ(α) represent for unimodular matrices?\",\"answer\":\"γ(α) is the maximum possible β=‖M−1‖ among unimodular integer matrices M with fixed entry bound ‖M‖=α. It measures how large entries of the inverse can get given a constraint on M.\"},{\"question\":\"How are the 3×3 submatrix A and g(A) used in the analysis?\",\"answer\":\"A is formed by deleting one row and one column of the 4×4 matrix M. For A, g(A) is defined as the gcd of its nine 2×2 minors, and it is used to control which determinant values are compatible with unimodularity.\"},{\"question\":\"What distinguishes the nonnegative and signed determinant bounds in the main theorem?\",\"answer\":\"The nonnegative case bounds |detA| using β+k, reflecting matrices with entries restricted to {0,…,k}. The signed case bounds |detA| using β with entries allowed in {−k,…,k}, and the paper emphasizes that the sharp extremal problems are genuinely distinct even though both bounds apply in overlapping situations.\"}]",1784200277,50,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"small-matrices-with-large-inverses-unimodular-44-cases","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/small-matrices-with-large-inverses-unimodular-44-cases/85014/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What does the parameter γ(α) represent for unimodular matrices?","Question",{"text":75,"@type":76},"γ(α) is the maximum possible β=‖M−1‖ among unimodular integer matrices M with fixed entry bound ‖M‖=α. It measures how large entries of the inverse can get given a constraint on M.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are the 3×3 submatrix A and g(A) used in the analysis?",{"text":80,"@type":76},"A is formed by deleting one row and one column of the 4×4 matrix M. For A, g(A) is defined as the gcd of its nine 2×2 minors, and it is used to control which determinant values are compatible with unimodularity.",{"name":82,"@type":73,"acceptedAnswer":83},"What distinguishes the nonnegative and signed determinant bounds in the main theorem?",{"text":84,"@type":76},"The nonnegative case bounds |detA| using β+k, reflecting matrices with entries restricted to {0,…,k}. The signed case bounds |detA| using β with entries allowed in {−k,…,k}, and the paper emphasizes that the sharp extremal problems are genuinely distinct even though both bounds apply in overlapping situations.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,114,119,122,126,129,133],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":28,"slug":113},6,"Technology","technology",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":21,"slug":125},9,"Religion & Spirituality","religion-spirituality",{"id":21,"doc_module":4,"doc_module_name":45,"category_name":127,"show_sort_weight":21,"slug":128},"World Cup","world-cup",{"id":130,"doc_module":4,"doc_module_name":45,"category_name":131,"show_sort_weight":130,"slug":132},10,"Lifestyle","lifestyle",{"id":134,"doc_module":4,"doc_module_name":45,"category_name":135,"show_sort_weight":106,"slug":136},19,"General","general"]