A note on the hit problem for the polynomial algebra in the case of odd primes and its application

TL;DR

This paper investigates the hit problem for polynomial algebras over finite fields at odd primes, providing new insights into algebraic transfer isomorphisms and the structure of Steenrod algebra modules.

Contribution

It offers a detailed analysis of $ ext{A}_p$-generators for polynomial algebras at odd primes and demonstrates the isomorphism of the third algebraic transfer in specific degrees.

Findings

The third algebraic transfer is an isomorphism in certain degrees.

Provides new $ ext{A}_p$-generator descriptions for polynomial algebras.

Enhances understanding of the $ ext{A}_p$-module structure at odd primes.

Abstract

Let $P_h = \mathbb{F}_p[t_1,\dots,t_h]$ be the polynomial algebra over $\mathbb{F}_p$ ($p$ prime). We consider the hit problem: finding a minimal generating set for $P_h$ as a module over the mod $p$ Steenrod algebra $\mathscr{A}_p$, or equivalently, determining a basis for $\mathbb{F}_p \otimes_{\mathscr{A}_p} P_h$. This problem is related to the $\mathscr{A}_p$-module structure of $H^*(V; \mathbb{F}_p) \cong \Lambda(V^\sharp) \otimes P_h$, where $V$ is an elementary abelian $p$-group of rank $h$. Information about the hit problem aids in studying the Singer algebraic transfer $Tr_h^{\mathscr{A}_p}$, a homomorphism from $GL(h, \mathbb{F}_p)$-coinvariants related to $H^*(V; \mathbb{F}_p)$ to ${\rm Ext}_{\mathscr{A}_p}^{h,h+*}(\mathbb{F}_p, \mathbb{F}_p)$, which helps analyze Ext groups. This work studies $\mathscr{A}_p$-generators for $P_h$ when $p$ is an odd prime. As an application, we investigate the third algebraic transfer ($h=3$) in certain generic degrees. Our main result shows that this transfer is an isomorphism in these degrees.