On the 퓐-generators of the polynomial algebra as a module over the Steenrod algebra, I.

Let 퓟_n := H^*((ℝP^∞)ⁿ) ≅ ℤ₂[x₁, x₂, ..., x_n] be the graded polynomial algebra over ℤ₂, where ℤ₂ denotes the prime field of two elements. We investigate the Peterson hit problem for the polynomial algebra 퓟_n, viewed as a graded left module over the mod-2 Steenrod algebra, 퓐. For n > 4, this problem is still unsolved, even in the case of n = 5 with the help of computers. In this article, we study the hit problem for the case n = 6 in the generic degree d_r = 6(2^r − 1) + 4.2^r with r an arbitrary non-negative integer. By considering ℤ₂ as a trivial 퓐-module, then the hit problem is equivalent to the problem of finding a basis of ℤ₂-vector space ℤ₂ ⊗_퓐퓟_n. The main goal of the current article is to explicitly determine an admissible monomial basis of the ℤ₂ vector space ℤ₂ ⊗_퓐퓟₆ in some degrees. As an application, the behavior of the sixth Singer algebraic transfer in the degree 6(2^r − 1) + 4.2^r is also discussed at the end of this paper. [ABSTRACT FROM AUTHOR]