On a minimal set of generators for the polynomial algebra of five variables as a module over the Steenrod algebra

TL;DR

This paper explicitly determines a minimal set of generators for the polynomial algebra in five variables over the Steenrod algebra, focusing on degrees of the form 4(2^d - 1), advancing understanding of the Peterson hit problem.

Contribution

It provides the first explicit minimal generating set for P_5 as an A-module in specific degrees, solving a case of the Peterson hit problem.

Findings

Explicit minimal generators for P_5 in degrees 4(2^d - 1).

Advances the solution to the Peterson hit problem for five variables.

Clarifies the structure of polynomial algebra modules over the Steenrod algebra.

Abstract

Denote by $P_k$ the graded polynomial algebra $\mathbb F_2[x_1,x_2,\ldots ,x_k]$ over the prime field of two elements, $\mathbb F_2$, with the degree of each $x_i$ being 1. We study the Peterson hit problem of determining a minimal set of generators for $P_k$ as a module over the mod-$2$ Steenrod algebra, $\mathcal{A}.$ In this paper, we explicitly determine a minimal set of $\mathcal{A}$-generators for $P_k$ in the case $k=5$ and the degree $4(2^d - 1)$ with $d$ an arbitrary positive integer.