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On the 𝓐-generators of the polynomial algebra as a module over the Steenrod algebra, I

  • Nguyen Khac Tin EMAIL logo , Phan Phuong Dung and Hoang Nguyen Ly
Published/Copyright: June 24, 2024
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 74 Issue 3

Abstract

Let π“Ÿn := H*((ℝP∞)n) β‰… β„€2[x1, x2, …, xn] be the graded polynomial algebra over β„€2, where β„€2 denotes the prime field of two elements. We investigate the Peterson hit problem for the polynomial algebra π“Ÿn, viewed as a graded left module over the mod-2 Steenrod algebra, 𝓐. For n > 4, this problem is still unsolved, even in the case of n = 5 with the help of computers. In this article, we study the hit problem for the case n = 6 in the generic degree dr = 6(2r βˆ’ 1) + 4.2r with r an arbitrary non-negative integer. By considering β„€2 as a trivial 𝓐-module, then the hit problem is equivalent to the problem of finding a basis of β„€2-vector space β„€2 βŠ—π“π“Ÿn. The main goal of the current article is to explicitly determine an admissible monomial basis of the β„€2 vector space β„€2 βŠ—π“π“Ÿ6 in some degrees. As an application, the behavior of the sixth Singer algebraic transfer in the degree 6(2r βˆ’ 1) + 4.2r is also discussed at the end of this paper.

MSC 2010: Primary 55S10; Secondary 55S05; 55T15
Keywords: Steenrod algebra; polynomial algebra; module and graded rings

This research is supported by Ho Chi Minh City University of Technology and Education (HCMUTE), Vietnam

Acknowledgement

The authors would like to express our warmest thanks to Prof. Nguyen Sum (Sai Gon University, Viet Nam) for helpful conversations. The first author thanks Mr. Dang Vo Phuc (Khanh Hoa University, Viet Nam) for a helpful discussion. Finally, the authors thank the referee for helpful comments.

  1. Communicated by Tibor Macko

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Received: 2022-08-06
Accepted: 2023-11-30
Published Online: 2024-06-24
Published in Print: 2024-06-25

Β© 2024 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2024-0058
Keywords for this article
Steenrod algebra; polynomial algebra; module and graded rings

Articles in the same Issue

  1. On the Paley graph of a quadratic character
  2. A topological duality for tense modal pseudocomplemented De Morgan algebras
  3. Fibonacci numbers as mixed concatenations of Fibonacci and Lucas numbers
  4. A general formula in composition theory
  5. A nonlinear Filbert-like matrix with three free parameters: From linearity to nonlinearity
  6. On universality in short intervals for zeta-functions of certain cusp forms
  7. On asymptotics for lacunary partition functions
  8. Parallel surfaces of the non-lightlike solution of vortex filament equations
  9. New q-analogues of Van Hamme’s (F.2) supercongruence and of some related supercongruences
  10. Kneser-type oscillation theorems for second-order functional differential equations with unbounded neutral coefficients
  11. Approximation theorems via Pp-statistical convergence on weighted spaces
  12. Global existence and multiplicity of positive solutions for anisotropic eigenvalue problems
  13. Theoretical analysis of higher-order system of difference equations with generalized balancing numbers
  14. On a solvable difference equations system of second order its solutions are related to a generalized Mersenne sequence
  15. Positive bases, cones, Helly-type theorems
  16. Digital Jordan surfaces arising from tetrahedral tiling
  17. Influence of ideals in compactifications
  18. On the functions Ο‰f and Ξ©f
  19. On the 𝓐-generators of the polynomial algebra as a module over the Steenrod algebra, I
  20. A bivariate distribution with generalized exponential conditionals
  21. A note on boundary feedback stabilization for degenerate parabolic equations in multi-dimensional domains
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