The hit problem for the polynomial algebra of four variables

TL;DR

This paper explicitly determines a minimal generating set of monomials for the polynomial algebra in four variables over the field with two elements as a module over the mod-2 Steenrod algebra, solving a specific algebraic problem.

Contribution

It provides an explicit solution to the hit problem for four variables, identifying the minimal generators in terms of monomials.

Findings

Explicit minimal generating set for four-variable case

Complete characterization of monomials as generators

Advances understanding of algebraic structure over Steenrod algebra

Abstract

We study the problem of determining a minimal set of generators for the polynomial algebra $\mathbb F_2[x_1,x_2,...,x_k]$ as a module over the mod-2 Steenrod algebra $\mathcal{A}$. In this paper, we give an explicit answer in terms of the monomials for $k=4$.