compute a classification of the complex roots of a polynomial system depending on parameters
ComplexRootClassification(F, d, R)
ComplexRootClassification(F, H, d, R)
ComplexRootClassification(CS, d, R)
list of polynomials
number of parameters
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.
R ≔ PolynomialRing⁡x,y,s
R ≔ polynomial_ring
F ≔ s−y+1⁢x,s−x+1⁢y
The computation below shows that the input parametric system can have 1 solution or 2 distinct solutions. The corresponding conditions on the parameters are given by constructible sets.
CC ≔ ComplexRootClassification⁡F,1,R
CC ≔ constructible_set,1,constructible_set,2
These constructible sets are printed below.
Download Help Document