  We study the hit problem, set up by F. Peterson, of finding a minimal set of
generators for the polynomial algebra $P_k := \mathbb F_2[x_1,x_2,...,x_k]$ as
a module over the mod-2 Steenrod algebra, $\mathcal{A}$. In this paper, we
study a minimal set of generators for $\mathcal A$-module $P_k$ in some so-call
generic degrees and apply these results to explicitly determine the hit problem
for $k=4$.
