The hit problem for the polynomial algebra of four variables

We study the problem of determining a minimal set of generators for the polynomial algebra ${\mathbb{F}}_2[x_1,x_2,\mathrm{.}\mathrm{.}\mathrm{.},x_k]$ as a module over the mod-2 Steenrod algebra $\mathcal{A}$. In this paper, we give an explicit answer in terms of the monomials for $k=4$.