Skip to main content
SpringerLink
Log in

A Note on Systems of Equations of Power Sums

  • Research Articles
  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

Symmetric functions play an important role in several subjects of mathematics, such as algebraic combinatorics, representation theory of finite groups and algebraic geometry. In this paper, we study the solutions of the following system of equations related to bases of symmetric functions:

$$\begin{cases} x_{1}^{k_1}+x_{2}^{k_1}+\cdots +x_{n}^{k_1}=a, \\ x_{1}^{k_2}+x_{2}^{k_2}+\cdots +x_{n}^{k_2}=a, \\ \vdots \\ x_{1}^{k_n}+x_{2}^{k_n}+\cdots +x_{n}^{k_n}=a, \end{cases}$$

where \(0<k_1<k_2<\cdots<k_n\), \(k_1,k_2,\cdots,k_n\) are natural numbers, and \(a\in\{0,1,n\}\). Our main theorems generalize several known results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D. Knuth, Sorting and Searching, in The Art of Computer Programming (Addison–Wesley, London, 1973), Vol. 3.

    MATH  Google Scholar 

  2. A. Lascoux, Symmetric Functions and Combinatorial Operators on Polynomials, in CBMS Regional Conference Series in Mathematics (American Mathematical Society, Providence, Rhode Island, 2003), Vol. 99.

    Book  Google Scholar 

  3. I. G. MacDonald, Symmetric functions and Hall polynomials (Oxford University Press, Oxford, 1979).

    MATH  Google Scholar 

  4. R. P. Stanley, Enumerative Combinatorics (Cambridge University Press, Cambridge, 1999), Vol. 2.

    Book  Google Scholar 

  5. E. R. Berlekamp, Algebraic Coding Theory (Aegean Park Press, Laguna Hills, CA, 1984).

    MATH  Google Scholar 

  6. J. A. Eidswick, “A proof of Newton’s power sum formulas,” Amer. Math. Monthly 75 (4), 396–397 (1968).

    Article  MathSciNet  Google Scholar 

  7. R. Zippel, Effective Polynomial Computation (Kluwer Academic Publishers, Boston, 1993).

    Book  Google Scholar 

  8. D. G. Mead, “Newton’s identities,” Amer. Math. Monthly 99 (8), 749–751 (1992).

    Article  MathSciNet  Google Scholar 

  9. D. E. Dobbs and R. Hanks, A Modern Course on the Theory of Equations, 2nd ed. (Polygonal Publishing House, Washington, New Jersey, 1992).

    MATH  Google Scholar 

  10. W. Wu, Mathematics Mechanization (Science Press, Beijing, 2003) [in Chinese].

    Google Scholar 

  11. X. Wang and S. Yang, “On solving equations of algebraic sum of equal powers,” Sci. China Ser. A 49 (9), 1153–1157 (2006).

    Article  MathSciNet  Google Scholar 

  12. Y. Wu and C. N. Hadjicostis, “On solving composite power polynomial equations,” Math. Comp. 74 (250), 853–868 (2004).

    Article  MathSciNet  Google Scholar 

  13. Y. Wu, “More on solving systems of power equations,” Math. Comp. 79 (272), 2317–2332 (2010).

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

We thank the editors and anonymous referees for their time and comments.

Funding

This work was supported by National Natural Science Foundation of China (grants no. 12171142, no. 11971155, no. 12071117).

Author information

Authors and Affiliations

  1. Hubei University, Wuhan, 430062, China

    R. Gao, J. Liao & X. Xu

  2. Hainan University, Haikou, 570228, China

    H. Liu

  3. Harbin Institute of Technology, Harbin, 150001, China

    H. Xiong

Authors
  1. R. Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Liao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. H. Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. H. Xiong
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. X. Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gao, R., Liao, J., Liu, H. et al. A Note on Systems of Equations of Power Sums. Math Notes 111, 855–869 (2022). https://doi.org/10.1134/S0001434622050194

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434622050194

Keywords

Search

Navigation