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DOI: 10.4230/LIPIcs.FSTTCS.2009.2327
URN: urn:nbn:de:0030-drops-23278
URL: http://drops.dagstuhl.de/opus/volltexte/2009/2327/

Kumar, Abhinav ; Lokam, Satyanarayana V. ; Patankar, Vijay M. ; Sarma M. N., Jayalal

### Using Elimination Theory to construct Rigid Matrices

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### Abstract

The rigidity of a matrix $A$ for target rank $r$ is the minimum number of entries of $A$ that must be changed to ensure that the rank of the altered matrix is at most $r$. Since its introduction by Valiant \cite{Val77}, rigidity and similar rank-robustness functions of matrices have found numerous applications in circuit complexity, communication complexity, and learning complexity. Almost all $\nbyn$ matrices over an infinite field have a rigidity of $(n-r)^2$. It is a long-standing open question to construct infinite families of \emph{explicit} matrices even with superlinear rigidity when $r=\Omega(n)$. In this paper, we construct an infinite family of complex matrices with the largest possible, i.e., $(n-r)^2$, rigidity. The entries of an $\nbyn$ matrix in this family are distinct primitive roots of unity of orders roughly \SL{$\exp(n^4 \log n)$}. To the best of our knowledge, this is the first family of concrete (but not entirely explicit) matrices having maximal rigidity and a succinct algebraic description. Our construction is based on elimination theory of polynomial ideals. In particular, we use results on the existence of polynomials in elimination ideals with effective degree upper bounds (effective Nullstellensatz). Using elementary algebraic geometry, we prove that the dimension of the affine variety of matrices of rigidity at most $k$ is exactly $n^2 - (n-r)^2 +k$. Finally, we use elimination theory to examine whether the rigidity function is semicontinuous.

### BibTeX - Entry

@InProceedings{kumar_et_al:LIPIcs:2009:2327,
author =	{Abhinav Kumar and Satyanarayana V. Lokam and Vijay M. Patankar and Jayalal Sarma M. N.},
title =	{Using Elimination Theory to construct Rigid Matrices},
booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2009)},
pages =	{299--310},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-939897-13-2},
ISSN =	{1868-8969},
year =	{2009},
volume =	{4},
editor =	{Ravi Kannan and K Narayan Kumar},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address =	{Dagstuhl, Germany},
URL =		{http://drops.dagstuhl.de/opus/volltexte/2009/2327},
URN =		{urn:nbn:de:0030-drops-23278},
doi =		{http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2009.2327},
annote =	{Keywords: Matrix Rigidity, Lower Bounds, Circuit Complexity}
}


 Keywords: Matrix Rigidity, Lower Bounds, Circuit Complexity Seminar: IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science Issue date: 2009 Date of publication: 14.12.2009

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