Zeon多项式的谱单零点
Spectrally Simple Zeros of Zeon Polynomials
摘要：
Zeon的组合性质已被应用于图计数问题、图着色、通信网络中的路由问题、划分依赖的随机积分和布尔可满足性。利用Zeon函数的幂零性质，初等Zeon函数的幂级数可以自然地化为有限和。此外，任何解析复变函数的Zeon扩展在Zeon的相关等价类上具有Zeon多项式表示。本文研究了复Zeon上多项式的零点。将已有的关于实Zeon多项式的结果推广到复情形，并建立了新的结果。特别地，建立了复Zeon多项式的谱单零点的Zeon代数的一个基本定理，并给出了当谱单零点存在时求谱单零点的算法。作为应用，使用多项式方法计算了解析函数的zeon延拓的逆。
英文摘要：
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.
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