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

Nguyen Sum, Pham Do Tai

TL;DR

The Peterson hit problem asks for a minimal $\mathbb{F}_2$-basis of $P_k$ as a module over the mod-$2$ Steenrod algebra $\mathcal{A}$. This paper explicitly solves the problem for $k=5$ in the generic degree $m=2^{d}$ with $d\ge8$, showing $\dim (QP_5)_{2^{d}}=1984$ and giving a concrete, admissible-monomial description yielding exactly 1984 generators. The authors develop and apply a weight-vector framework, leveraging spike theory, Singer’s results, and the $p_{(i;I)}$ maps to decompose $QP_5$ into weight-by-weight components, then enumerate and prove independence of the admissible monomials in each weight class. Appendices provide explicit monomial lists and the structure of low-degree spaces, clarifying the full generating set and the SF-generated relations. The work advances understanding of the hit problem and has implications for representations of general linear groups via the action of the Steenrod algebra.

Abstract

Let $P_k$ be the graded polynomial algebra $\mathbb F_2[x_1,x_2,\ldots ,x_k]$ 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 $d \geqslant 8$.

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

TL;DR

The Peterson hit problem asks for a minimal -basis of as a module over the mod- Steenrod algebra . This paper explicitly solves the problem for in the generic degree with , showing and giving a concrete, admissible-monomial description yielding exactly 1984 generators. The authors develop and apply a weight-vector framework, leveraging spike theory, Singer’s results, and the maps to decompose into weight-by-weight components, then enumerate and prove independence of the admissible monomials in each weight class. Appendices provide explicit monomial lists and the structure of low-degree spaces, clarifying the full generating set and the SF-generated relations. The work advances understanding of the hit problem and has implications for representations of general linear groups via the action of the Steenrod algebra.

Abstract

Let be the graded polynomial algebra over the prime field with two elements, , with the degree of each being 1. We study the hit problem, set up by Frank Peterson, of finding a minimal set of generators for as a module over the mod- Steenrod algebra, It is an open problem in Algebraic Topology. In this paper, we explicitly determine a minimal set of -generators for in the case of the generic degree for all .
Paper Structure (21 sections, 27 theorems, 106 equations)

This paper contains 21 sections, 27 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The indecomposables of $P_5$ in degree $2^{d}$
  4. The weight vector of admissible monomials of degree $2^d$
  5. Computation of $QP_5((2)|(1)|^{d-1})$
  6. Computation of $QP_5^+((4)|(2)|^{d-2})$ for $d \geqslant 6$
  7. Computation of $QP_5((4)|^2|(3)|^{d-4}|(1))$
  8. Appendix 1
  9. The admissible monomials of weight vector $(4)|(2)|^{d-2}$ in $P_5$
  10. The admissible monomials of weight vector $(4)|^2|(3)|^{d-4}$ in $P_5$
  11. The admissible monomials of weight vector $(4)|^2|(3)|^{d-4}|(1)$ in $P_5$
  12. Appendix 2
  13. A note on the structure of the space $(QP_5)_{32}$
  14. Computation of $(QP_5)_{64}$
  15. Computation of $QP_5((4)|^2(3)|^2|(1))$
  16. ...and 6 more sections

Key Result

Theorem 1.1

For any integer $d \geqslant 8$, there exist exactly $1984$ admissible monomials of degree $2^{d}$ in $P_5$. Consequently $\dim (QP_5)_{2^{d}} = 1984.$

Theorems & Definitions (46)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: See Kameko ka, Sum su1
  • Proposition 2.6: See su2
  • Definition 2.7
  • Theorem 2.8: See Singer si2
  • Lemma 2.9: Phúc-Sum sp
  • ...and 36 more