论文标题
用于参数多项式系统的最终工具，并应用于人群模型
Resultant Tools for Parametric Polynomial Systems with Application to Population Models
论文作者
论文摘要
我们关注的是分解多项式方程参数系统的参数空间，以及可能与该系统达到的实际解决方案的数量有关的一些多项式不等式约束。先前的研究对此问题采用了两步方法，在该方法中，首先通过Groebner（GB）计算系统的判别品种，然后将其圆柱代数分解（CAD）计算出来，以提供所需的计算。但是，即使在一些相当小的应用示例中，这个过程太昂贵了，单独使用判别品种的计算是不可行的。在本文中，我们开发了使用结果方法（使用迭代单变量结果的新方法和一种新方法）开发新的方法来构建判别品种。与GB相比，这降低了复杂性，并可以解决以前的不可行的例子。我们通过对人口动态的问题提供符号解决方案来证明益处 - 对三个相连的人群的稳态分析，这些稳态表现出表现的效果 - 以前只能在数值上解决。
We are concerned with the problem of decomposing the parameter space of a parametric system of polynomial equations, and possibly some polynomial inequality constraints, with respect to the number of real solutions that the system attains. Previous studies apply a two step approach to this problem, where first the discriminant variety of the system is computed via a Groebner Basis (GB), and then a Cylindrical Algebraic Decomposition (CAD) of this is produced to give the desired computation. However, even on some reasonably small applied examples this process is too expensive, with computation of the discriminant variety alone infeasible. In this paper we develop new approaches to build the discriminant variety using resultant methods (the Dixon resultant and a new method using iterated univariate resultants). This reduces the complexity compared to GB and allows for a previous infeasible example to be tackled. We demonstrate the benefit by giving a symbolic solution to a problem from population dynamics -- the analysis of the steady states of three connected populations which exhibit Allee effects - which previously could only be tackled numerically.
