The command RepresentingRootIndex(pbx,R) returns the representing root index in a parametric box.

See the page SemiAlgebraicSetTools for the definitions of a regular semi-algebraic set, a parametric box and a box.

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

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

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

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

$R≔\mathrm{polynomial\_ring}$

$F≔\left[x^2+d\,a{d}^2+bd+c\right]$

$F≔\left[x^2+d\,a{d}^2+bd+c\right]$

$N≔\left[\right]$

$N≔\left[\right]$

$P≔\left[\right]$

$P≔\left[\right]$

$H≔\left[\right]$

$H≔\left[\right]$

$\mathrm{rrc}≔\mathrm{RealRootClassification}\left(F\,N\,P\,H\,4\,2\,R\right)$

$\mathrm{rrc}≔\left[\left[\mathrm{regular\_semi\_algebraic\_set}\right]\,\mathrm{border\_polynomial}\right]$

$\mathrm{rsas}≔\mathrm{rrc}\left[1\right]\left[1\right]$

$\mathrm{rsas}≔\mathrm{regular\_semi\_algebraic\_set}$

$\mathrm{rbx}≔\mathrm{RepresentingBox}\left(\mathrm{rsas}\,R\right)$

$\mathrm{rbx}≔\mathrm{parametric\_box}$

$\mathrm{DisplayParametricBox}\left(\mathrm{rbx}\right)\:$

$d<0$

$\mathrm{AND}$

$c_1\&colon;d^{2}a+db+c\&equals;0$

$\mathrm{RepresentingRootIndex}\left(\mathrm{rbx}\right)$

$\left[\left[1\right]\right]$