Resultant Tools for Parametric Polynomial Systems with Application to Population Models

TL;DR

This paper introduces new resultant-based methods to decompose parameter spaces of polynomial systems, enabling symbolic analysis of population models that were previously only solvable numerically.

Contribution

It develops novel approaches using Dixon and iterated univariate resultants to efficiently compute discriminant varieties, improving over traditional Gröbner basis methods.

Findings

Reduced computational complexity for discriminant variety computation

Enabled symbolic analysis of complex population models

Successfully applied to steady states analysis in population dynamics

Abstract

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.