Abstract
Zeon algebras arise as commutative subalgebras of fermions, and can be constructed as subalgebras of Clifford algebras of appropriate signature. Their combinatorial properties have been applied to graph enumeration problems, stochastic integrals, and even routing problems in communication networks. Analogous to real polynomial functions, zeon polynomial functions are defined as zeon-valued polynomial functions of a zeon variable. In this paper, properties of zeon polynomials and their zeros are considered. Nilpotent and invertible zeon zeros of polynomials with real coefficients are characterized, and necessary conditions are established for the existence of zeros of polynomials with zeon coefficients. Quadratic polynomials with zeon coefficients are considered in detail. A “zeon quadratic formula” is developed, and solutions of \(ax^2+bx+c=0\) are characterized with respect to the “zeon discriminant” of the equation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ben Slimane, J., Schott, R., Song, Y-Q., Staples, G.S., Tsiontsiou, E.: Operator calculus algorithms for multi-constrained paths. Int. J. Math. Comput. Sci. 10, 69–104 (2015). http://ijmcs.future-in-tech.net/10.1/R-Jamila.pdf. Accessed 20 Jan 2019
Cruz-Sánchez, H., Staples, G.S., Schott, R., Song, Y-Q.: Operator calculus approach to minimal paths: Precomputed routing in a store-and-forward satellite constellation. In: Proceedings of IEEE Globecom 2012, Anaheim, CA, USA, December 3–7, 3455–3460. ISBN: 978-1-4673-0919-6
Dollar, L.M., Staples, G.S.: Zeon roots. Adv. Appl. Clifford Algebras 27, 1133–1145 (2017). https://doi.org/10.1007/s00006-016-0732-4
Nefzi, B., Schott, R., Song, Y.-Q., Staples, G.S., Tsiontsiou, E.: An operator calculus approach for multi-constrained routing in wireless sensor networks, ACM MobiHoc, Hangzhou. China (to appear) (2015)
Neto, A.F.: Higher order derivatives of trigonometric functions, Stirling numbers of the second kind, and zeon algebra. J. Integer Seq. 17, 14 (2014)
Neto, A.F.: Carlitz’s identity for the Bernoulli numbers and zeon algebra. J. Integer Seq. 18, 15 (2015)
Neto, A.F.: A note on a theorem of Guo, Mezö, and Qi. J. Integer Seq. 19, 16 (2016)
Neto, A.F., dos Anjos, P.H.R.: Zeon algebra and combinatorial identities. SIAM Rev. 56, 353–370 (2014)
Schott, R., Staples, G.S.: Partitions and Clifford algebras. Eur. J. Combin. 29, 1133–1138 (2008)
Schott, R., Staples, G.S.: Zeons, lattices of partitions, and free probability. Comm. Stoch. Anal. 4, 311–334 (2010)
Staples, G.S.: CliffMath: Clifford algebra computations in Mathematica, 2008–2018. http://www.siue.edu/~sstaple/index_files/research.htm. Accessed 20 Jan 2019
Staples, G.S.: A new adjacency matrix for finite graphs. Adv. Appl. Clifford Algebras 18, 979–991 (2008)
Staples, G.S.: Hamiltonian cycle enumeration via fermion-zeon convolution. Int. J. Theor. Phys. 56, 3923–3934 (2017)
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
Staples, G.S., Weygandt, A.: Elementary functions and factorizations of zeons. Adv. Appl. Clifford Algebras 28, 12 (2018)
Acknowledgements
The authors thank the anonymous reviewers for their careful reading and helpful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Eckhard Hitzer
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Haake, E., Staples, G.S. Zeros of Zeon Polynomials and the Zeon Quadratic Formula. Adv. Appl. Clifford Algebras 29, 21 (2019). https://doi.org/10.1007/s00006-019-0938-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00006-019-0938-3