Geodesic
Zero-nonzero patterns for nilpotent matrices over finite fields
The electronic journal of linear algebra, Tome 18 (2009), pp. 628-648.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: Fix a field F. A zero-nonzero pattern A is said to be potentiallynilpotent over F if there exists a matrix with entries in F with zero-nonzero pattern A that allows nilpotence. In this paper an investigation is initiated into which zero-nonzero patterns are potentiallynilpotent over F with a special emphasis on the case that F = Z p is a finite field. A necessarycondition on F is observed for a pattern to be potentiallynilpotent when the associated digraph has m loops but no small k-cycles, $2 \leq k \leq m - 1$. As part of this investigation, methods are developed, using the tools of algebraic geometryand commutative algebra, to eliminate zero-nonzero patterns A as being potentiallynilpotent over anyfield F. These techniques are then used to classifyall irreducible zero-nonzero patterns of order two and three that are potentiallynilpotent over Zp for each prime p.
Classification : 15A18, 13P10, 05C50, 11T06
Keywords: zero-nonzero patterns, nilpotent, ideal saturation, gr$\ddot $obner basis, finite fields 
@article{ELA_2009__18__a11,
     author = {Vander Meulen, Kevin N. and Van Tuyl, Adam},
     title = {Zero-nonzero patterns for nilpotent matrices over finite fields},
     journal = {The electronic journal of linear algebra},
     pages = {628--648},
     publisher = {mathdoc},
     volume = {18},
     year = {2009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ELA_2009__18__a11/}
}
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Vander Meulen, Kevin N.; Van Tuyl, Adam. Zero-nonzero patterns for nilpotent matrices over finite fields. The electronic journal of linear algebra, Tome 18 (2009), pp. 628-648. http://geodesic.mathdoc.fr/item/ELA_2009__18__a11/