{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article6640","name":"Using Elimination Theory to construct Rigid Matrices","abstract":"The rigidity of a matrix $A$ for target rank $r$ is the minimum number\r\nof entries of $A$ that must be changed to ensure that the rank of\r\nthe altered matrix is at most $r$. Since its introduction by Valiant\r\n\\cite{Val77}, rigidity and similar rank-robustness functions of\r\nmatrices have found numerous applications in circuit complexity,\r\ncommunication complexity, and learning complexity. Almost all $\\nbyn$\r\nmatrices over an infinite field have a rigidity of $(n-r)^2$. It is a\r\nlong-standing open question to construct infinite families of\r\n\\emph{explicit} matrices even with superlinear rigidity when $r=\\Omega(n)$.\r\n\r\nIn this paper, we construct an infinite family of complex matrices\r\nwith the largest possible, i.e., $(n-r)^2$, rigidity. The entries of\r\nan $\\nbyn$ matrix in this family are distinct primitive roots of unity\r\nof orders roughly \\SL{$\\exp(n^4 \\log n)$}. To the best of our knowledge, this is\r\nthe first family of concrete (but not entirely explicit) matrices\r\nhaving maximal rigidity and a succinct algebraic description.\r\n\r\nOur construction is based on elimination theory of polynomial\r\nideals. In particular, we use results on the existence of polynomials\r\nin elimination ideals with effective degree upper bounds (effective\r\nNullstellensatz). Using elementary algebraic geometry, we prove that\r\nthe dimension of the affine variety of matrices of rigidity at\r\nmost $k$ is exactly $n^2 - (n-r)^2 +k$. Finally, we use elimination theory to\r\nexamine whether the rigidity function is semicontinuous.","keywords":["Matrix Rigidity","Lower Bounds","Circuit Complexity"],"author":[{"@type":"Person","name":"Kumar, Abhinav","givenName":"Abhinav","familyName":"Kumar"},{"@type":"Person","name":"Lokam, Satyanarayana V.","givenName":"Satyanarayana V.","familyName":"Lokam"},{"@type":"Person","name":"Patankar, Vijay M.","givenName":"Vijay M.","familyName":"Patankar"},{"@type":"Person","name":"Sarma M. N., Jayalal","givenName":"Jayalal","familyName":"Sarma M. N."}],"position":26,"pageStart":299,"pageEnd":310,"dateCreated":"2009-12-14","datePublished":"2009-12-14","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Kumar, Abhinav","givenName":"Abhinav","familyName":"Kumar"},{"@type":"Person","name":"Lokam, Satyanarayana V.","givenName":"Satyanarayana V.","familyName":"Lokam"},{"@type":"Person","name":"Patankar, Vijay M.","givenName":"Vijay M.","familyName":"Patankar"},{"@type":"Person","name":"Sarma M. N., Jayalal","givenName":"Jayalal","familyName":"Sarma M. N."}],"copyrightYear":"2009","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.FSTTCS.2009.2327","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6207","volumeNumber":4,"name":"IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science","dateCreated":"2009-12-07","datePublished":"2009-12-07","editor":[{"@type":"Person","name":"Kannan, Ravi","givenName":"Ravi","familyName":"Kannan"},{"@type":"Person","name":"Narayan Kumar, K.","givenName":"K.","familyName":"Narayan Kumar"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article6640","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6207"}}}