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We are concerned with the problem of decomposing the parameter space of aparametric system of polynomial equations, and possibly some polynomialinequality constraints, with respect to the number of real solutions that thesystem attains. Previous studies apply a two step approach to this problem,where first the discriminant variety of the system is computed via a GroebnerBasis (GB), and then a Cylindrical Algebraic Decomposition (CAD) of this isproduced to give the desired computation. However, even on some reasonablysmall applied examples this process is too expensive, with computation of thediscriminant variety alone infeasible. In this paper we develop new approachesto build the discriminant variety using resultant methods (the Dixon resultantand a new method using iterated univariate resultants). This reduces thecomplexity compared to GB and allows for a previous infeasible example to betackled. We demonstrate the benefit by giving a symbolic solution to a problemfrom population dynamics -- the analysis of the steady states of threeconnected populations which exhibit Allee effects - which previously could onlybe tackled numerically.

Resultant Tools for Parametric Polynomial Systems with Application to Population Models