Zeros of Zeon Polynomials and the Zeon Quadratic Formula

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.