参数多项式系统的结式工具及其在数学生物学中的应用
Resultant Tools for Parametric Polynomial Systems with Application to Mathematical Biology
AI总结：
本文提出利用Dixon结式和迭代单变量结式等结式方法构建参数多项式系统的判别簇，以替代Gröbner基方法，并在数学生物学示例中展示其能处理更大规模问题。
摘要：
为了根据参数多项式方程组所达到的实数解的数量来分解该方程组的参数空间，首先需要计算该系统的判别簇。这是一个消元类型的问题，因为判别簇是原始系统簇与一个或多个新超曲面的交集在参数空间中的投影的并集。尽管Gröbner基（GBs）在此任务中很流行，但在本工作中，我们关注一种较少使用的替代工具，即结式。我们开发了使用不同结式方法（如Dixon结式和迭代单变量结式）构建判别簇的新方法。我们讨论了最小判别簇、我们方法计算出的判别簇以及已知的边界多项式之间的关系。我们通过种群动力学和化学反应网络理论中的众多示例证明，我们的新方法能够成功处理比GB所能处理的更大规模的示例。
英文摘要：
To decompose the parameter space of a parametric system of polynomial equations with respect to the number of real solutions that the system attains, first the discriminant variety of the system should be computed. This is an elimination type problem because the discriminant variety is the union of the projections into the parameter space of the intersection of the variety of the original system with one or several new hypersurfaces. Despite the popularity of Gröbner Bases (GBs) for this task, in this work we focus on a lesser used alternative tool, namely resultants. We develop new approaches to build the discriminant variety using different resultant methods such as Dixon resultant and iterated univariate resultants. We discuss how the minimal discriminant variety, the discriminant varieties computed by our methods, and the known border polynomials are related. We demonstrate with numerous examples from population dynamics and chemical reaction network theory that our new approach successfully handles larger size examples than a GB can.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
