On the Peterson hit problem

Abstract

We study the hit problem, set up by F. Peterson, of finding a minimal set of generators for the polynomial algebra $P_k:={\mathbb{F}}_2[x_1,x_2,...,x_k]$ as a module over the mod-2 Steenrod algebra, $\mathcal{A}$. In this paper, we study a minimal set of generators for $\mathcal{A}$-module $P_k$ in some so-call generic degrees and apply these results to explicitly determine the hit problem for $k=4$.

68 pages, Quy Nhon University preprint, Viet Nam, 2011. A shorter version of this paper was published in Advances in Mathematics

References in corpus (3)

• Sub-Hopf algebras of the Steenrod algebra and the Singer transfer
• The hit problem for the polynomial algebra of four variables
• Young tableaux and the Steenrod algebra

Cited by in corpus (19)

• On the generators of the polynomial algebra as a module over the Steenrod algebra
• The "hit" problem of five variables in the generic degree and its application
• On a construction for the generators of the polynomial algebra as a module over the Steenrod algebra
• On a minimal set of generators for the polynomial algebra of five variables as a module over the Steenrod algebra
• The affirmative answer to Singer's conjecture on the algebraic transfer of rank four
• On the determination of the Singer transfer
• 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
• A note on the hit problem for the polynomial algebra in the case of odd primes and its application
• On the lambda algebra and Singer's cohomological transfer
• On Singer's conjecture for the fourth algebraic transfer in certain generic degrees
• A note on the hit problem for the Steenrod algebra and its applications
• The squaring operartion and the Singer algebraic transfer
• On the algebraic transfers of ranks 4 and 6 at generic degrees
• On Singer's conjecture for the fifth algebraic transfer
• The admissible monomial bases for the polynomial algebra of five variables in some types of generic degrees
• A note on the modular representation on the $\mathbb{Z}/2$-homology groups of the fourth power of real projective space and its application
• On the hit problem for the Steenrod algebra in some generic degrees and applications
• Structure of the space of $G{L}_4({\mathbb{Z}}_2)$-coinvariants ${\mathbb{Z}}_2{\otimes }_{G{L}_4({\mathbb{Z}}_2)}P{H}_{\ast }({\mathbb{Z}}_2^4,{\mathbb{Z}}_2)$ in some generic degrees and its application