The squaring operation and the hit problem for the polynomial algebra in a type of generic degree

TL;DR

This paper investigates the kernel of Kameko's squaring operation in the polynomial algebra over , providing explicit solutions to the hit problem for five variables in a specific degree, advancing understanding of algebraic structures under Steenrod operations.

Contribution

It introduces a generating set for the kernel of Kameko's squaring operation in a generic degree and applies this to solve the hit problem for five variables.

Findings

Explicit kernel generating set for the squaring operation.

Solved the hit problem for P_5 in a generic degree.

Enhanced understanding of module structures over the Steenrod algebra.

Abstract

Let $P_k$ be the graded polynomial algebra $\mathbb F_2[x_1,x_2,\ldots ,x_k]$ with the degree of each generator $x_i$ being 1, where $\mathbb F_2$ denote the prime field with two elements. The hit problem of Frank Peterson asks for a minimal generating set for the polynomial algebra $P_k$ as a module over the mod-2 Steenrod algebra $\mathcal{A}$. Equivalently, we want to find a vector space basis for $\mathbb F_2 \otimes_{\mathcal A} P_k$ in each degree. In this paper, we study a generating set for the kernel of Kameko's squaring operation $\widetilde{Sq}^0_*: \mathbb F_2 \otimes_{\mathcal A} P_k \longrightarrow \mathbb F_2 \otimes_{\mathcal A} P_k$ in a so-called generic degree. By using this result, we explicitly compute the hit problem for $k=5$ in the respective generic degree.