A minimal set of generators for the polynomial algebra of five variables in a generic degree

Abstract

Let $P_k$ be the graded polynomial algebra ${\mathbb{F}}_2[x_1,x_2,\dots ,x_k]$ over the prime field with 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, $\mathcal{A}.$ It is an open problem in Algebraic Topology. In this paper, we explicitly determine a minimal set of $\mathcal{A}$-generators for $P_5$ in the case of the generic degree $m=2^d$ for all $d⩾8$.

60 pages. arXiv admin note: substantial text overlap with arXiv:2209.12543; text overlap with arXiv:1609.02250 by other authors