The integer d must be positive and smaller than the number of variables.

The characteristic of R must be zero and the last d variables of R are regarded as parameters.

For a parametric algebraic system, this command computes all the possible numbers of solutions of this system together with the corresponding necessary and sufficient conditions on its parameters.

More precisely, let V be the variety defined by F. The command ComplexRootClassification(F, d, R) returns a classification of the complex roots of F depending on parameters, that is, a finite partition P of the parameter space into constructible sets such that above each part, the number of solutions of V is either infinite or constant.

If a constructible set CS is specified, the representing regular systems of CS must be square-free. The function call ComplexRootClassification(CS, d, R) returns a classification of the points of the constructible set CS, that is, a finite partition P of the parameter space into constructible sets such that above each part, the number of solutions of CS is either infinite or constant.

If H is specified, let $W$ be the variety defined by the product of polynomials in H. The command ComplexRootClassification(F, H, d, R) returns a classification of the points of the constructible set V-W depending on parameters.

$\mathrm{with}\left(\mathrm{RegularChains}\right)\:$

$\mathrm{with}\left(\mathrm{ConstructibleSetTools}\right)\:$

$\mathrm{with}\left(\mathrm{ParametricSystemTools}\right)\:$

$R≔\mathrm{PolynomialRing}\left(\left[x\,y\,s\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$F≔\left[s-\left(y+1\right)x\,s-\left(x+1\right)y\right]$

$F≔\left[s-\left(y+1\right)x\,s-\left(x+1\right)y\right]$

$\mathrm{CC}≔\mathrm{ComplexRootClassification}\left(F\,1\,R\right)$

$\mathrm{CC}≔\left[\left[\mathrm{constructible\_set}\,1\right]\,\left[\mathrm{constructible\_set}\,2\right]\right]$

$\mathrm{map}\left(x\mapsto \left[\mathrm{Info}\left(x\left[1\right]\,R\right)\,x\left[2\right]\right]\,\mathrm{CC}\right)$

$\left[\left[\left[\left[4s+1\right]\,\left[1\right]\right]\,1\right]\,\left[\left[\left[\right]\,\left[s\,4s+1\right]\right]\,\left[\left[s\right]\,\left[1\right]\right]\,2\right]\right]$