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](..).

$\mathrm{with}\left(\mathrm{RegularChains}\right)\:$

$\mathrm{with}\left(\mathrm{ConstructibleSetTools}\right)\:$

$\mathrm{with}\left(\mathrm{ParametricSystemTools}\right)\:$

$R≔\mathrm{PolynomialRing}\left(\left[x\,a\,b\,c\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$F≔a{x}^2+bx+c$

$F≔a{x}^2+bx+c$

$\mathrm{ds}≔\mathrm{DiscriminantSet}\left(\left[F\right]\,3\,R\right)$

$\mathrm{ds}≔\mathrm{constructible\_set}$

$\mathrm{ds}≔\mathrm{MakePairwiseDisjoint}\left(\mathrm{ds}\,R\right)$

$\mathrm{ds}≔\mathrm{constructible\_set}$

$\mathrm{Info}\left(\mathrm{ds}\,R\right)$

$\left[\left[a\,b\right]\,\left[1\right]\right]$