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

Abstract

Let $P_s:={\mathbb{F}}_2[x_1,x_2,\dots ,x_s]$ be the graded polynomial algebra over the prime field of two elements, ${\mathbb{F}}_2$, in $s$ variables $x_1,x_2,\dots ,x_s$, each of degree one. This algebra is considered as a graded module over the mod-2 Steenrod algebra, $\mathcal{A}$. We are interested in the "hit" problem of finding a minimal set of generators for $\mathcal{A}$-module $P_s.$ This problem is unresolved for every $s⩾5.$ 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.

29 pages. Comments very welcome!

References in corpus (5)

• On the generators of the polynomial algebra as a module over the Steenrod algebra
• Sub-Hopf algebras of the Steenrod algebra and the Singer transfer
• On the determination of the Singer transfer
• Young tableaux and the Steenrod algebra
• On the lambda algebra and Singer's cohomological transfer

Cited by in corpus (8)

• The affirmative answer to Singer's conjecture on the algebraic transfer of rank four
• On the dimension of $H^{\ast }(({\mathbb{Z}}_2{)}^{\times t},{\mathbb{Z}}_2)$ as a module over Steenrod ring
• On the lambda algebra and Singer's cohomological transfer
• A note on the non-trivial elements in the cohomology groups of the Steenrod algebra
• 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