The "hit" problem of five variables in the generic degree and its application

Let $P_s:={\mathbb{F}}_2[x_1,x_2,\dots ,x_s]$ be the graded polynomial algebra over the prime field of two elements, ${\mathbb{F}}_2$, in $s$ variables $x_1, x_2, \ldots , x_s$, each of degree one. This algebra is considered as a graded module over the mod-2 Steenrod algebra, $\mathcal{A}$. We are interested in the “hit” problem of finding a minimal set of generators for $\mathscr A$-module$P_s.$Thisproblemisunresolvedforevery$s\geqslant 5.$ In this paper, we study the hit problem of five variables in a generic degree, from which we investigate Singer’s conjecture [Math. Z. 202 (1989), 493-523] for the transfer homomorphism of rank $5$ in degrees given. This gives an efficient method to study the algebraic transfer and it is different from the ones of Singer.