compute the discriminant set of a variety
DiscriminantSet(F, d, R)
list of polynomials
number of parameters
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](..).
R ≔ PolynomialRing⁡x,a,b,c
Consider the following general quadratic polynomial F.
F ≔ a⁢x2+b⁢x+c
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.
ds ≔ DiscriminantSet⁡F,3,R
ds ≔ MakePairwiseDisjoint⁡ds,R
The first case indicates that there are infinite number of solutions; the second one indicates that there is no solution.
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