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

Let $P_k$ be the graded polynomial algebra ${\mathbb{F}}_2[x_1,x_2,\dots ,x_k]$ 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 ${\widetilde{Sq}}_{\ast }^0:{\mathbb{F}}_2{\otimes }_{\mathcal{A}}{P}_k⟶{\mathbb{F}}_2{\otimes }_{\mathcal{A}}{P}_k$ in a so-called generic degree. By using this result, we explicitly compute the hit problem for $k=5$ in the respective generic degree.