Spectrally Simple Zeros of Zeon Polynomials

TL;DR

This paper extends the theory of zeros of zeon polynomials to complex zeons, establishing a fundamental theorem for spectrally simple zeros and providing an algorithm for their computation, with applications to inverse functions.

Contribution

It introduces a fundamental theorem for spectrally simple zeros of complex zeon polynomials and presents an algorithm for finding these zeros, expanding zeon algebra theory.

Findings

Extended real zeon polynomial results to complex zeons.

Established a fundamental theorem for spectrally simple zeros.

Developed an algorithm to compute spectrally simple zeros.

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.