The "hit" problem of five variables in the generic degree and its application

TL;DR

This paper investigates the hit problem for five variables in a generic degree within the polynomial algebra over _2, providing new insights into Singer's conjecture and offering an alternative method to study algebraic transfer.

Contribution

It introduces a novel approach to the hit problem for five variables and explores its implications for Singer's conjecture on algebraic transfer.

Findings

New method for studying the hit problem in five variables.

Insights into Singer's conjecture for rank 5.

Alternative approach to algebraic transfer analysis.

Abstract

Let $P_s:= \mathbb F_2[x_1,x_2,\ldots ,x_s]$ be the graded polynomial algebra over the prime field of two elements, $\mathbb F_2$, in $s$ variables $x_1, x_2, \ldots , x_s$, each of degree one. This algebra is considered as a graded module over the mod-2 Steenrod algebra, $\mathscr {A}$. We are interested in the "hit" problem of finding a minimal set of generators for $\mathscr A$-module $P_s.$ This problem is unresolved for every $s\geqslant 5.$ In this paper, we study the hit problem of five variables in a generic degree, from which we investigate Singer's conjecture [Math. Z. 202 (1989), 493-523] for the transfer homomorphism of rank $5$ in degrees given. This gives an efficient method to study the algebraic transfer and it is different from the ones of Singer.