RepresentingRootIndex - Maple Help

RegularChains[SemiAlgebraicSetTools]

 RepresentingRootIndex
 return the root index in a parametric box

 Calling Sequence RepresentingRootIndex(pbx,R)

Parameters

 pbx - parametric box R - polynomial ring

Description

 • 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.

Examples

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

Determine under which conditions on $a,b,c,d$ the equation ${x}^{2}+d=0$ has $2$ distinct real solutions provided that $d$ satisfies $a{d}^{2}+bd+c=0$.

 > $F≔\left[{x}^{2}+d,a{d}^{2}+bd+c\right]$
 ${F}{≔}\left[{{x}}^{{2}}{+}{d}{,}{a}{}{{d}}^{{2}}{+}{b}{}{d}{+}{c}\right]$ (2)
 > $N≔\left[\right]$
 ${N}{≔}\left[\right]$ (3)
 > $P≔\left[\right]$
 ${P}{≔}\left[\right]$ (4)
 > $H≔\left[\right]$
 ${H}{≔}\left[\right]$ (5)
 > $\mathrm{rrc}≔\mathrm{RealRootClassification}\left(F,N,P,H,4,2,R\right)$
 ${\mathrm{rrc}}{≔}\left[\left[{\mathrm{regular_semi_algebraic_set}}\right]{,}{\mathrm{border_polynomial}}\right]$ (6)
 > $\mathrm{rsas}≔\mathrm{rrc}\left[1\right]\left[1\right]$
 ${\mathrm{rsas}}{≔}{\mathrm{regular_semi_algebraic_set}}$ (7)
 > $\mathrm{rbx}≔\mathrm{RepresentingBox}\left(\mathrm{rsas},R\right)$
 ${\mathrm{rbx}}{≔}{\mathrm{parametric_box}}$ (8)
 > $\mathrm{DisplayParametricBox}\left(\mathrm{rbx}\right):$
 ${d}{<}{0}$
 ${\mathrm{AND}}$
 ${{c}}_{{1}}{:}{{d}}^{{2}}{}{a}{+}{d}{}{b}{+}{c}{=}{0}$ (9)
 > $\mathrm{RepresentingRootIndex}\left(\mathrm{rbx}\right)$
 $\left[\left[{1}\right]\right]$ (10)