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  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.