Skip to main content
Log in

Spectrally Simple Zeros of Zeon Polynomials

  • Published:
Advances in Applied Clifford Algebras Aims and scope Submit manuscript

Abstract

Combinatorial properties of zeons have been applied to graph enumeration problems, graph colorings, routing problems in communication networks, partition-dependent stochastic integrals, and Boolean satisfiability. Power series of elementary zeon functions are naturally reduced to finite sums by virtue of the nilpotent properties of zeons. Further, the zeon extension of any analytic complex function has zeon polynomial representations on associated equivalence classes of zeons. In this paper, zeros of polynomials over complex zeons are considered. Existing results for real zeon polynomials are extended to the complex case and new results are established. In particular, a fundamental theorem of zeon algebra is established for spectrally simple zeros of complex zeon polynomials, and an algorithm is presented that allows one to find spectrally simple zeros when they exist. As an application, inverses of zeon extensions of analytic functions are computed using polynomial methods.

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

Access this article

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

Lisa M. Dollar & G. Stacey Staples

Notes

  1. The name “zeon algebra” was coined by Feinsilver [3], stressing their relationship to both bosons (commuting generators) and fermions (null-square generators).

  2. The n-particle (real) zeon algebra has also been denoted by \({\mathcal {C}\ell _n}^\mathrm{nil}\) in a number of papers because it can be constructed as a subalgebra of the Clifford algebra \(\mathcal {C}\ell _{n,n}\).

  3. The term “dual” here is motivated by regarding zeons as higher-dimensional dual numbers.

  4. In particular, \(\kappa \) is the least positive integer such that \((\mathfrak {D}u)^\kappa =0\).

  5. The zeon extension \(\varphi _{f,n}\) is the polynomial function obtained by extending the domain of f from \(\mathbb {C}\) to \(\mathbb {C}\mathfrak {Z}_n\).

  6. Among other properties, squares of zeon elements must be of even grade.

References

  1. Davis, A., Staples, G.S.: Zeon and idem-Clifford formulations of Boolean satisfiability. Adv. Appl. Clifford Algebras 29, 60 (2019). https://doi.org/10.1007/s00006-019-0978-8

    Article  MathSciNet  MATH  Google Scholar 

  2. Dollar, L.M., Staples, G.S.: Zeon roots. Adv. Appl. Clifford Algebra 27, 1133–1145 (2017). https://doi.org/10.1007/s00006-016-0732-4

  3. Feinsilver, P.: Zeon algebra, Fock space, and Markov chains. Commun. Stoch. Anal. 2, 263–275 (2008)

    MathSciNet  MATH  Google Scholar 

  4. Feinsilver, P., McSorley, J.: Zeons, permanents, the Johnson scheme, and generalized derangements. Int. J. Comb. Article ID 539030, pp. 29 (2011). https://doi.org/10.1155/2011/539030

  5. Haake, E., Staples, G.S.: Zeros of zeon polynomials and the zeon quadratic formula. Adv. Appl. Clifford Algebras 29, 21 (2019)

    Article  MathSciNet  Google Scholar 

  6. Mansour, T., Schork, M.: On the Differential Equation of First and Second Order in the Zeon Algebra. Adv. Appl. Clifford Algebras 31, 21 (2021). https://doi.org/10.1007/s00006-021-01126-7

    Article  MathSciNet  MATH  Google Scholar 

  7. Neto, A.F.: Higher order derivatives of trigonometric functions. Stirling numbers of the second kind, and zeon algebra. J. Integer Seq. 17, Article 14.9.3 (2014)

  8. Neto, A.F.: Carlitz’s identity for the Bernoulli numbers and zeon algebra. J. Integer Seq. 18, Article 15.5.6 (2015)

  9. Neto, A.F.: A note on a theorem of Guo, Mezö, and Qi. J. Integer Seq. 19, Article 16.4.8 (2016)

  10. Neto, A.F., dos Anjos, P.H.R.: Zeon algebra and combinatorial identities. SIAM Rev. 56, 353–370 (2014)

    Article  MathSciNet  Google Scholar 

  11. Schott, R., Staples, G.S.: Operator Calculus on Graphs (Theory and Applications in Computer Science). Imperial College Press, London (2012)

    Book  Google Scholar 

  12. Staples, G.S., Stellhorn, T.: Zeons, orthozeons, and graph colorings. Adv. Appl. Clifford Algebras 27, 1825–1845 (2017). https://doi.org/10.1007/s00006-016-0732-4

    Article  MathSciNet  MATH  Google Scholar 

  13. Staples, G.S.: Clifford Algebras and Zeons: Geometry to Combinatorics and Beyond. World Scientific Publishing (2019). ISBN 978-981-120-257-5

  14. Staples, G.S.: Differential calculus of zeon functions. Adv. Appl. Clifford Algebras 29, 25 (2019)

    Article  MathSciNet  Google Scholar 

  15. Staples, G.S.: CliffMath: Clifford algebra computations in Mathematica. Available for download at http://www.siue.edu/~sstaple/index_files/research.html

  16. Staples, G.S.: Zeon matrix inverses and the zeon combinatorial Laplacian. Adv. Appl. Clifford Algebras 31, 40 (2021). https://doi.org/10.1007/s00006-021-01152-5

    Article  MathSciNet  MATH  Google Scholar 

  17. Staples, G.S., Weygandt, A.: Elementary Functions and Factorizations of Zeons. Adv. Appl. Clifford Algebras 28, 12 (2018)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author thanks the anonymous referees for their comments, suggestions, and careful reading of the manuscript.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to G. Stacey Staples.

Additional information

Communicated by Rafał Abłamowicz

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Staples, G.S. Spectrally Simple Zeros of Zeon Polynomials. Adv. Appl. Clifford Algebras 31, 66 (2021). https://doi.org/10.1007/s00006-021-01167-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00006-021-01167-y

Keywords

Mathematics Subject Classification

Navigation