The admissible monomial bases for the polynomial algebra of five variables in some types of generic degrees

Abstract

Let $P_k$ be the graded polynomial algebra ${\mathbb{F}}_2[x_1,x_2,\dots ,x_k]$ over the prime field of two elements, ${\mathbb{F}}_2$, with the degree of each $x_i$ being 1. We study the hit problem, set up by Frank Peterson, of finding a minimal set of generators for $P_k$ as a module over the mod-$2$ Steenrod algebra, $\mathcal{A}.$ In this paper, we explicitly determine a minimal set of $\mathcal{A}$-generators for $P_5$ in the case of the degrees $n=2^{d+1}-1$ and $n=2^{d+1}-2$ for all $d⩾6$.

35 pages. arXiv admin note: text overlap with arXiv:1609.02250, arXiv:2103.04393 by other authors

References in corpus (3)

• On the generators of the polynomial algebra as a module over the Steenrod algebra
• On the dimension of $H^{\ast }(({\mathbb{Z}}_2{)}^{\times t},{\mathbb{Z}}_2)$ as a module over Steenrod ring
• The squaring operation and the hit problem for the polynomial algebra in a type of generic degree