Higherdistance commuting varieties
Abstract
The commuting variety of matrices over a given field is a wellstudied object in linear algebra and algebraic geometry. As a set, it consists of all pairs of square matrices with entries in that field that commute with one another. In this paper we generalise the commuting variety by using the commuting distance of matrices. We show that over an algebraically closed field, each of our sets does indeed form a variety. We compute the dimension of the distance$2$ commuting variety and characterize its irreducible components. We also work over other fields, showing that the distance$2$ commuting set is a variety but that the higher distance commuting sets may or may not be varieties, depending on the field and on the size of the matrices.
 Publication:

arXiv eprints
 Pub Date:
 November 2018
 arXiv:
 arXiv:1811.09553
 Bibcode:
 2018arXiv181109553E
 Keywords:

 Mathematics  Algebraic Geometry;
 14M12;
 15A27
 EPrint:
 21 pages