On the Peterson hit problem

TL;DR

This paper investigates the hit problem, aiming to find minimal generators for polynomial algebras over the Steenrod algebra, and explicitly solves it for four variables in certain degrees.

Contribution

It provides a new analysis of minimal generators in generic degrees and explicitly solves the hit problem for four variables.

Findings

Determined minimal generators for specific degrees in polynomial algebras.

Explicitly solved the hit problem for k=4 in certain degrees.

Enhanced understanding of the structure of polynomial algebras over the Steenrod algebra.

Abstract

We study the hit problem, set up by F. Peterson, of finding a minimal set of generators for the polynomial algebra $P_k := \mathbb F_2[x_1,x_2,...,x_k]$ as a module over the mod-2 Steenrod algebra, $\mathcal{A}$. In this paper, we study a minimal set of generators for $\mathcal A$-module $P_k$ in some so-call generic degrees and apply these results to explicitly determine the hit problem for $k=4$.