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.

zeon 的组合特性已应用于图枚举问题、图着色、通信网络中的路由问题、依赖分区的随机积分和布尔可满足性。凭借 zeon 的幂零特性，基本 zeon 函数的幂级数自然会减少到有限和。此外，任何解析复函数的 zeon 扩展在相关的 zeon 等价类上具有 zeon 多项式表示。在本文中，考虑了复数 zeon 上多项式的零点。实数 zeon 多项式的现有结果扩展到复数情况，并建立新的结果。特别是，对于复 zeon 多项式的谱简单零点，建立了 zeon 代数的基本定理，并且提出了一种算法，当它们存在时，它允许人们找到谱简单的零点。作为一种应用，使用多项式方法计算解析函数的 zeon 扩展的逆。