On the generators of the polynomial algebra as a module over the Steenrod algebra

TL;DR

This paper investigates the minimal generating set of polynomial algebras over the field of two elements as modules over the Steenrod algebra, focusing on specific degrees related to powers of two.

Contribution

It provides new results on the hit problem in certain degrees, extending previous work by Mothebe.

Findings

Determined minimal generators in degree (k-1)(2^d-1)

Extended the understanding of the hit problem for polynomial algebras

Connected results to previous research by Mothebe

Abstract

Let $P_k:= \mathbb F_2[x_1,x_2,\ldots,x_k]$ be the polynomial algebra over the prime field of two elements, $\mathbb F_2$, in $k$ variables $x_1, x_2, \ldots, x_k$, each of degree 1. We are interested in the Peterson hit problem of finding a minimal set of generators for $P_k$ as a module over the mod-2 Steenrod algebra, $\mathcal{A}$. In this paper, we study the hit problem in degree $(k-1)(2^d-1)$ with $d$ a positive integer. Our result implies the one of Mothebe [4,5].