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

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.