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 ${\mathcal{A}}_p$, or equivalently, determining a basis for ${\mathbb{F}}_p{\otimes }_{{\mathcal{A}}_p}{P}_h$. This problem is related to the ${\mathcal{A}}_p$-module structure of $H^{\ast }(V;{\mathbb{F}}_p)\cong \mathrm{\Lambda }(V^{\mathrm{♯}})\otimes P_h$, where $V$ is an elementary abelian $p$-group of rank $h$. Information about the hit problem aids in studying the Singer algebraic transfer $T{r}_h^{{\mathcal{A}}_p}$, a homomorphism from $GL(h,{\mathbb{F}}_p)$-coinvariants related to $H^{\ast }(V;{\mathbb{F}}_p)$ to ${\mathrm{E}\mathrm{x}\mathrm{t}}_{{\mathcal{A}}_p}^{h,h+\ast }({\mathbb{F}}_p,{\mathbb{F}}_p)$, which helps analyze Ext groups. This work studies ${\mathcal{A}}_p$-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.