RegularChains[ParametricSystemTools] - Maple Help

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RegularChains[ParametricSystemTools]

 DiscriminantSet
 compute the discriminant set of a variety

 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

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$
 > $\mathrm{with}\left(\mathrm{ConstructibleSetTools}\right):$
 > $\mathrm{with}\left(\mathrm{ParametricSystemTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[x,a,b,c\right]\right)$
 ${R}{:=}{\mathrm{polynomial_ring}}$ (1)

Consider the following general quadratic polynomial F.

 > $F≔a{x}^{2}+bx+c$
 ${F}{:=}{a}{}{{x}}^{{2}}{+}{b}{}{x}{+}{c}$ (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.

 > $\mathrm{ds}≔\mathrm{DiscriminantSet}\left(\left[F\right],3,R\right)$
 ${\mathrm{ds}}{:=}{\mathrm{constructible_set}}$ (3)
 > $\mathrm{ds}≔\mathrm{MakePairwiseDisjoint}\left(\mathrm{ds},R\right)$
 ${\mathrm{ds}}{:=}{\mathrm{constructible_set}}$ (4)
 > $\mathrm{Info}\left(\mathrm{ds},R\right)$
 $\left[\left[{a}{,}{b}\right]{,}\left[{1}\right]\right]$ (5)

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