compute a classification of the complex roots of a polynomial system depending on parameters - Maple Help

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RegularChains[ParametricSystemTools][ComplexRootClassification] - compute a classification of the complex roots of a polynomial system depending on parameters

Calling Sequence

ComplexRootClassification(F, d, R)

ComplexRootClassification(F, H, d, R)

ComplexRootClassification(CS, d, R)

Parameters

F

-

list of polynomials

H

-

list of polynomials

d

-

number of parameters

R

-

polynomial ring

CS

-

constructible set

Description

• 

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.

Examples

withRegularChains:

withConstructibleSetTools:

withParametricSystemTools:

R:=PolynomialRingx,y,s

R:=polynomial_ring

(1)

F:=sy+1x,sx+1y

F:=sy+1x,sx+1y

(2)

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:=ComplexRootClassificationF,1,R

CC:=constructible_set,1,constructible_set,2

(3)

These constructible sets are printed below.

mapx→Infox1,R,x2,CC

4s+1,1,1,,s,4s+1,s,1,2

(4)

See Also

ComprehensiveTriangularize, ConstructibleSetTools, ParametricSystemTools, RealRootClassification, RegularChains


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