A note on the generators of the polynomial algebra of six variables and application

"Let $\mathcal P_{n}:=H^{*}((\mathbb{R}P^{\infty})^{n}) \cong \mathbb Z_2[x_{1},x_{2},\ldots,x_{n}]$ be the graded polynomial algebra over $\mathcal K,$ where $\mathcal K$ denotes the prime field of two elements. We investigate the Peterson hit problem for the polynomial algebra $\mathcal P_{n},$ viewed as a graded left module over the mod-$2$ Steenrod algebra, $\mathcal{A}.$ For $n>4,$ this problem is still unsolved, even in the case of $n=5$ with the help of computers. In this paper, we study the hit problem for the case $n=6$ in degree $d_{k}=6(2^{k} -1)+9.2^{k},$ with $k$ an arbitrary non-negative integer. By considering $\mathcal K$ as a trivial $\mathcal A$-module, then the hit problem is equivalent to the problem of finding a basis of $\mathcal K$-graded vector space $\mathcal K {\otimes}_{\mathcal{A}}\mathcal P_{n}.$ The main goal of the current paper is to explicitly determine an admissible monomial basis of the $\mathcal K$-graded vector space $\mathcal K{\otimes}_{\mathcal{A}}\mathcal P_6$ in some degrees. At the same time, the behavior of the sixth Singer algebraic transfer in degree $d_{k}=6(2^{k} -1)+9.2^{k}$ is also discussed at the end of this article. Here, the Singer algebraic transfer is a homomorphism from the homology of the mod-$2$ Steenrod algebra, $\mbox{Tor}^{\mathcal{A}}_{n, n+d}(\mathcal K, \mathcal K),$ to the subspace of $\mathcal K\otimes_{\mathcal{A}}\mathcal P_{n}$ consisting of all the $GL_n$-invariant classes of degree $d.$"