AI 中文总结
研究工程完全交这类稀疏多项式系统，通过推广混合细分概念热带化系统，设计热带同伦延拓算法，给出计算消元式牛顿多胞形的算法，实现软件包展示算法实用性。
AI 中文摘要
工程完全交（ECI）是一类稀疏多项式系统，在纯数学（如枚举几何）和应用（如化学反应网络）中经常出现。基于第一作者给出的理论结果，我们有多项贡献。首先，通过将Huber和Sturmfels（1995）引入的混合细分的经典概念推广到ECI，给出一种新的有效技术来热带化此类系统，特别旨在有效计算ECI形式的方阵方程组的解。进一步设计了一种热带同伦延拓算法来计算这种混合细分。我们的技术可通过与Helminck、Henriksson和Ren（2024）引入的算法结合来数值求解此类系统。最后，给出一种计算ECI消元式的牛顿多胞形的算法，这为计算例如所谓的A判别式的牛顿多胞形提供了新方法。我们以软件包形式实现算法，并在一系列示例上展示其实用可行性。
英文摘要
Engineered Complete Intersections (ECI's) are a class of sparse polynomial systems frequently arising in a number of contexts, both in pure mathematics (e.g. enumerative geometry) and applications (e.g. chemical reaction networks). Based on theoretical results given by the first author, we give several contributions. First we give a new effective technique to tropicalize such systems by generalizing the classical notion of mixed subdivisions introduced by Huber and Sturmfels (1995) to ECI's with the particular goal to efficiently count solutions of square systems of equations in ECI form. We further design a tropical homotopy continuation algorithm for computing such mixed subdivisions, inspired by Jensen (2016), Malajovich (2017) and Daisey and Ren (2024). Our techniques can be used to numerically solve such systems by coupling them with the algorithms introduced by Helminck, Henriksson and Ren (2024). Finally, we give an algorithm to compute Newton polytopes of eliminants of ECI's. This gives a new way to compute, for example, Newton polytopes of so-called $A$-discriminants. Coupled with evaluation-interpolation paradigms our algorithm gives an efficient approach to compute such eliminants. We implemented our algorithms in the form of a software package which we use to demonstrate their practical feasibility on a range of examples.
Comments34 pages, 1 figure
