DIMENSIONS OF SUBSPACES OF THE POLYNOMIAL ALGEBRA  GENERATED BY SPIKES

Mbakiso Mothebe (Botswana)

Received September 28, 2007

Abstract

Let  and  be integers. Let  be the polynomial algebra in n-variables  over the field  of two elements and let  be the mod-2 Steenrod algebra. In this paper we give a formula for computing  the number of degree d monomials of the form  (called spikes) in  Motivation for this question comes from algebraic topology where  is identified with the mod-2 cohomology group of the n-fold product of  with itself and thereby receives its module structure over  The value of  is of interest in the problem of determining a basis for the quotient vector space  of the polynomial algebra by the image of the positive part  of the Steenrod algebra [1, 11, 13].

Keywords and phrases: dimension, spike, polynomial algebra.