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RegularChains[ParametricSystemTools]

  

DiscriminantSet

  

compute the discriminant set of a variety

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

DiscriminantSet(F, d, R)

Parameters

F

-

list of polynomials

d

-

number of parameters

R

-

polynomial ring

Description

• 

The command DiscriminantSet(F, d, R) returns the discriminant set of a polynomial system with respect to a positive integer, which is a constructible set.

• 

d is positive and less than the number of variables in R.

• 

Given a positive integer d, the last d variables will be regarded as parameters.

• 

A point P is in the discriminant set of F if and only if after specializing F at P, the polynomial system F has no solution or an infinite number of solutions.

• 

This command is part of the RegularChains[ParametricSystemTools] package, so it can be used in the form DiscriminantSet(..) only after executing the command with(RegularChains[ParametricSystemTools]). However, it can always be accessed through the long form of the command by using RegularChains[ParametricSystemTools][DiscriminantSet](..).

Examples

withRegularChains:

withConstructibleSetTools:

withParametricSystemTools:

RPolynomialRingx,a,b,c

R:=polynomial_ring

(1)

Consider the following general quadratic polynomial F.

Fax2+bx+c

F:=ax2+bx+c

(2)

You can see that when F as a univariate polynomial in x has no solution (over the complex number field) or has infinitely many number solutions.

dsDiscriminantSetF,3,R

ds:=constructible_set

(3)

dsMakePairwiseDisjointds,R

ds:=constructible_set

(4)

Infods,R

a,b,1

(5)

The first case indicates that there are infinite number of solutions; the second one indicates that there is no solution.

See Also

ComprehensiveTriangularize

ConstructibleSet

DefiningSet

Info

ParametricSystemTools

PreComprehensiveTriangularize

RegularChains

Triangularize

 


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