The admissible monomial bases for the polynomial algebra of five variables in some types of generic degrees

TL;DR

This paper explicitly determines minimal sets of generators for the polynomial algebra in five variables over F_2 as a module over the Steenrod algebra, focusing on specific degrees related to powers of two minus one or two.

Contribution

It provides explicit minimal generators for P_5 over the Steenrod algebra in degrees of the form 2^{d+1}-1 and 2^{d+1}-2 for all d ≥ 6, advancing the hit problem.

Findings

Explicit minimal generators for P_5 in specified degrees.

Progress in solving the hit problem for five-variable polynomial algebras.

Clarification of the structure of P_5 as an A-module in these degrees.

Abstract

Let $P_k$ be 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 hit problem, set up by Frank Peterson, 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 explicitly determine a minimal set of $\mathcal{A}$-generators for $P_5$ in the case of the degrees $n = 2^{d+1} - 1$ and $n = 2^{d+1} - 2$ for all $d \geqslant 6$.