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International Journal of Mathematics and Mathematical Sciences
Volume 2015, Article ID 235806, 7 pages
http://dx.doi.org/10.1155/2015/235806
Research Article

Some Relations between Admissible Monomials for the Polynomial Algebra

1Department of Mathematics, University of Botswana, Private Bag 00704, Gaborone, Botswana
2African Institute for Mathematical Sciences, 6 Melrose Road, Muizenberg, Cape Town, South Africa

Received 14 April 2015; Accepted 5 July 2015

Academic Editor: Ram N. Mohapatra

Copyright © 2015 Mbakiso Fix Mothebe and Lafras Uys. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let be the polynomial algebra in variables , of degree one, over the field of two elements. The mod-2 Steenrod algebra acts on according to well known rules. A major problem in algebraic topology is of determining , the image of the action of the positively graded part of . We are interested in the related problem of determining a basis for the quotient vector space . has been explicitly calculated for but problems remain for . Both and are graded, where denotes the set of homogeneous polynomials of degree . In this paper, we show that if is an admissible monomial (i.e., meets a criterion to be in a certain basis for ), then, for any pair of integers (), , and , the monomial is admissible. As an application we consider a few cases when .