Zeros of cubic polynomials in zeon algebra

Bach Do; G. Stacey Staples

doi:10.56754/0719-0646.2703.553

DOI:

https://doi.org/10.56754/0719-0646.2703.553

Abstract

It is well known that every cubic polynomial with complex coefficients has three not necessarily distinct complex zeros. In this paper, zeros of cubic polynomials over complex zeons are considered. In particular, a monic cubic polynomial with zeon coefficients may have three spectrally simple zeros, uncountably many zeros, or no zeros at all. A classification of zeros is developed based on an extension of the cubic discriminant to zeon polynomials. In indeterminate cases, sufficient conditions are provided for existence of spectrally nonsimple zeon zeros. We also show that when considering zeros of cubic polynomials over the finite-dimensional complex zeon algebra \(\mathcal{C}\mathcal{3}_2\), there are no indeterminate cases.

Keywords

Zeons , polynomials , cubic formula , symbolic computation

Mathematics Subject Classification:

13B25 , 05E40 , 81R05

Bach Do Department of Mathematics & Statistics. Southern Illinois University Edwardsville, Edwardsville, IL 62026-1653, USA. https://orcid.org/0009-0000-0234-7110
G. Stacey Staples Department of Mathematics & Statistics. Southern Illinois University Edwardsville, Edwardsville, IL 62026-1653, USA. https://orcid.org/0000-0003-4755-7423

Pages: 553–579
Date Published: 2025-11-21
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