RegularChains[SemiAlgebraicSetTools] - Maple Programming Help

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RegularChains[SemiAlgebraicSetTools]

 RepresentingBox
 return the representing box of a semi-algebraic set

 Calling Sequence RepresentingBox(rst, R)

Parameters

 rst - regular semi-algebraic set R - polynomial ring

Description

 • The command RepresentingBox(rst, R) returns the representing box or parametric box of a regular semi-algebraic set.
 • 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)
 > $\mathrm{rsas}≔{{\mathrm{rrc}}_{1}}_{1}$
 ${\mathrm{rsas}}{:=}{\mathrm{regular_semi_algebraic_set}}$ (7)
 > $\mathrm{rbx}≔\mathrm{RepresentingBox}\left(\mathrm{rsas},R\right)$
 ${\mathrm{rbx}}{:=}{\mathrm{parametric_box}}$ (8)
 > $\mathrm{Info}\left(\mathrm{rbx},R\right)$
 $\left[\left[\left[{d}\right]{,}\left[\left[{-}{1}\right]\right]\right]{,}\left[{a}{}{{d}}^{{2}}{+}{b}{}{d}{+}{c}\right]{,}\left[\left[{1}\right]\right]\right]$ (9)