The hit problem of five variables in the generic degree $5(2^{s}-1) + 42.2^{s}$ and its application
We denote by $\mathbb Z_2$ the prime field of two elements and by $P_t = \mathbb Z_2[x_1, \ldots, x_t]$ the polynomial algebra of $t$ generators $x_1, \ldots, x_t$ with $\deg(x_j) = 1.$ Let $\mathcal A_2$ be the Steenrod algebra over $\mathbb Z_2.$ A central problem of homotopy theory is to determine a minimal set of generators for the $\mathbb Z_2$-graded vector space $\{(\mathbb Z_2\otimes_{\mathcal A_2} P_t)_n\}_{n\geq 0}.$ It is called \textit{the "hit" problem} for Steenrod algebra and has been completely solved for $t\leq 4.$ In this article, we explicitly solve the hit problem of five variables in the generic degree $5(2^{s}-1) + 42.2^{s}$ for any $s\geq 0.$ The result confirms Sum's conjecture \cite{N.S2} for the relation between the minimal sets of $\mathcal A_2$-generators of the algebras $P_{t-1}$ and $P_{t}$ in the case $t=5$ and the above generic degree. An efficient approach to surveying the hit problem of five variables has been presented. As an application, we obtain the dimension of $(\mathbb Z_2\otimes_{\mathcal A_2} P_t)_n$ for $t = 6$ and the generic degree $n = 5(2^{s+5}-1) + 42.2^{s+5}$ for all $s\geq 0.$ At the same time, we show that the fifth Singer algebraic transfer is an isomorphism in bidegree $(5, 47.2^{s})$ with $s\geq 0.$