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.
翻译:我们担心的是,多元等式参数系统参数空间的分解问题,以及可能存在一些多边不平等限制的问题,这涉及到这个系统实现的实际解决办法的数量。以前的研究对这个问题采用了两步方法,首先通过格罗布纳基准(GB)来计算这个系统的差异多样性,然后产生一个这一系统的圆柱形藻变形(CAD)来进行预期的计算。然而,即使在一些相当小的应用例子中,这个过程也太昂贵,只计算相异的种类是行不通的。在本文件中,我们制定了新的方法,利用由此产生的方法(狄克逊结果和使用循环的未流生结果的新方法)来构建这种差异多样性。这降低了这个系统的复杂性,并使得可以用以前不可行的例子来解决这个问题。我们通过给人口动态问题提供象征性的解决方案来证明它的好处,即分析显示Allee效应的三个相关人口的稳定状态,这些特征过去只能用数字方法解决。