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On a Construction for the Generators of the Polynomial Algebra as a Module Over the Steenrod Algebra

  • Nguyễn SumEmail author
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

Let \(P_n\) be the graded polynomial algebra \(\mathbb F_2[x_1,x_2,\ldots ,x_n]\) with the degree of each generator \(x_i\) being 1, where \(\mathbb F_2\) denote the prime field of two elements. The Peterson hit problem is to find a minimal generating set for \(P_n\) regarded as a module over the mod-2 Steenrod algebra, \(\mathcal {A}\). Equivalently, we want to find a vector space basis for \(\mathbb F_2 \otimes _{\mathcal A} P_n\) in each degree d. Such a basis may be represented by a list of monomials of degree d. In this paper, we present a construction for the \(\mathcal A\)-generators of \(P_n\) and prove some properties of it. We also explicitly determine a basis of \(\mathbb F_2 \otimes _{\mathcal A} P_n\) for \(n = 5\) and the degree \(d = 15.2^s - 5\) with s an arbitrary positive integer. These results are used to verify Singer’s conjecture for the fifth Singer algebraic transfer in respective degree.

Keywords

Steenrod algebra Peterson hit problem Algebraic transfer Polynomial algebra 

1991 Mathematics Subject Classification

Primary 55S10 Secondary 55S05 55T15 

Notes

Acknowledgements

The first manuscript of this paper was written when the author was visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM) from August to November 2017. He would like to thank the VIASM for financial support, convenient working condition and kind hospitality. I would like to thank the referee for helpful comments and suggestions which have led to the improvement of the paper’s exposition. The author was supported in part by the National Foundation for Science and Technology Development (NAFOSTED) of Vietnam under the grant number 101.04-2017.05.

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Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Application, Sài Gòn UniversityHo chí Minh cityVietnam

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