The command SolveTools[SemiAlgebraic] computes the solutions of a rational polynomial system consisting of a combination of equations, inequations, and inequalities over the real numbers.

If vars is not specified, the command determines the list of unknowns to be solved.

SemiAlgebraic will solve for all the unknown symbols in the input system. If vars is specified, but does not include all symbols in the system then the omitted symbols will be treated as parameters and the output will be a piecewise expression. The parameters option can be used to specify an ordered list of parameters.

The order of equations and inequalities in the solutions is determined by the order of the variables given in vars.  If vars is not specified, the order that was used to solve the polynomial system is used.

This command is part of the SolveTools package, and can be used in the form SemiAlgebraic(..) only after executing the command with(SolveTools).  However, it can always be accessed through the long form of the command using SolveTools[SemiAlgebraic](..).

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

$\mathrm{ineqs}≔\left\{0<3y-x-2\&comma;x^2+y^2<9\&comma;x^2-y<0\right\}$

$\mathrm{ineqs}≔\left\{0<3y-x-2\&comma;x^2-y<0\&comma;x^2+y^2<9\right\}$

$\mathrm{plots}:-\mathrm{inequal}\left(\mathrm{ineqs}\&comma;x=-2..2\&comma;y=0..3\&comma;\mathrm{color}=''Nautical\; Dark\; Gray''\right)$

$\mathrm{sasolxy}≔\mathrm{SolveTools}:-\mathrm{SemiAlgebraic}\left(\mathrm{ineqs}\&comma;\left[x\&comma;y\right]\right)$

$\mathrm{sasolxy}≔\left[\left[\mathrm{RootOf}\left({\mathrm{\_Z}}^4+{\mathrm{\_Z}}^2-9\&comma;\mathrm{-1.602}..\mathrm{-1.593}\right)<x\&comma;x<-\frac{2}{3}\&comma;x^2<y\&comma;y<\sqrt{-x^2+9}\right]\&comma;\left[x=-\frac{2}{3}\&comma;\frac{4}{9}<y\&comma;y<\frac{\sqrt{77}}{3}\right]\&comma;\left[-\frac{2}{3}<x\&comma;x<1\&comma;\frac{x}{3}+\frac{2}{3}<y\&comma;y<\sqrt{-x^2+9}\right]\&comma;\left[x=1\&comma;1<y\&comma;y<2\sqrt{2}\right]\&comma;\left[1<x\&comma;x<\mathrm{RootOf}\left({\mathrm{\_Z}}^4+{\mathrm{\_Z}}^2-9\&comma;1.593..1.602\right)\&comma;x^2<y\&comma;y<\sqrt{-x^2+9}\right]\right]$

$\mathrm{plots}:-\mathrm{inequal}\left(\mathrm{sasolxy}\&comma;x=-2..2\&comma;y=0..3\&comma;\mathrm{optionsfeasible}=\left[\mathrm{seq}\left(\left[\mathrm{color}=\mathrm{cat}\left(''Nautical''\&comma;\mathrm{floor}\left(\frac{i+1}{2}\right)\right)\right]\&comma;i=1..\mathrm{nops}\left(\mathrm{sasolxy}\right)\right)\right]\right)$

$\mathrm{sasolyx}≔\mathrm{SolveTools}:-\mathrm{SemiAlgebraic}\left(\mathrm{ineqs}\&comma;\left[y\&comma;x\right]\right)$

$\mathrm{sasolyx}≔\left[\left[\frac{4}{9}<y\&comma;y\le 1\&comma;-\sqrt{y}<x\&comma;x<3y-2\right]\&comma;\left[1<y\&comma;y\le -\frac{1}{2}+\frac{\sqrt{37}}{2}\&comma;-\sqrt{y}<x\&comma;x<\sqrt{y}\right]\&comma;\left[-\frac{1}{2}+\frac{\sqrt{37}}{2}<y\&comma;y<3\&comma;-\sqrt{-y^2+9}<x\&comma;x<\sqrt{-y^2+9}\right]\right]$

$\mathrm{plots}:-\mathrm{inequal}\left(\mathrm{sasolyx}\&comma;x=-2..2\&comma;y=0..3\&comma;\mathrm{optionsfeasible}=\left[\mathrm{seq}\left(\left[\mathrm{color}=\mathrm{cat}\left(''Nautical''\&comma;\mathrm{floor}\left(\frac{i+1}{2}\right)\right)\right]\&comma;i=1..\mathrm{nops}\left(\mathrm{sasolyx}\right)\right)\right]\right)$

$\mathrm{SemiAlgebraic}\left(\left\{x+y^2<0\&comma;x^2+x-1=0\right\}\right)$

$\left[\left[x=-\frac{\sqrt{5}}{2}-\frac{1}{2}\&comma;-\frac{\sqrt{2+2\sqrt{5}}}{2}<y\&comma;y<\frac{\sqrt{2+2\sqrt{5}}}{2}\right]\right]$

$\mathrm{SemiAlgebraic}\left(\left\{x+y^2<0\&comma;x^2+x-1=0\right\}\&comma;\left[y\&comma;x\right]\right)$

$\left[\left[\mathrm{RootOf}\left({\mathrm{\_Z}}^4-{\mathrm{\_Z}}^2-1\&comma;\mathrm{-1.274}..\mathrm{-1.265}\right)<y\&comma;y<\mathrm{RootOf}\left({\mathrm{\_Z}}^4-{\mathrm{\_Z}}^2-1\&comma;1.265..1.274\right)\&comma;x=-\frac{\sqrt{5}}{2}-\frac{1}{2}\right]\right]$

$\mathrm{SemiAlgebraic}\left(\left\{ax<b\right\}\&comma;\left[x\right]\right)$

$\left\{\begin{array}{cc}\left[\left[x=x\right]\right]& a=0\wedge 0<b\\ \left[\left[x<\frac{b}{a}\right]\right]& 0<a\\ \left[\left[\frac{b}{a}<x\right]\right]& a<0\\ \left[\right]& \mathrm{otherwise}\end{array}\right.$

$\mathrm{SemiAlgebraic}\left(\left\{ax<b\right\}\&comma;\left[x\right]\&comma;'\mathrm{parameters}'=\left[b\&comma;a\right]\right)$

$\left\{\begin{array}{cc}\left[\left[x<0\right]\right]& b=0\wedge 0<a\\ \left[\left[0<x\right]\right]& b=0\wedge a<0\\ \left[\left[x=x\right]\right]& 0<b\wedge a=0\\ \left[\left[x<\frac{b}{a}\right]\right]& 0<b\wedge 0<a\\ \left[\left[\frac{b}{a}<x\right]\right]& 0<b\wedge a<0\\ \left[\left[x<\frac{b}{a}\right]\right]& b<0\wedge 0<a\\ \left[\left[\frac{b}{a}<x\right]\right]& b<0\wedge a<0\\ \left[\right]& \mathrm{otherwise}\end{array}\right.$

$\mathrm{SemiAlgebraic}\left(\left\{x+y^2<a\&comma;x^2+x-1=0\right\}\&comma;\left[x\&comma;y\right]\right)$

$\left\{\begin{array}{cc}\left[\right]& a\le -\frac{\sqrt{5}}{2}-\frac{1}{2}\\ \left[\left[x=-\frac{\sqrt{5}}{2}-\frac{1}{2}\&comma;-\frac{\sqrt{2\sqrt{5}+2+4a}}{2}<y\&comma;y<\frac{\sqrt{2\sqrt{5}+2+4a}}{2}\right]\right]& a\le \frac{\sqrt{5}}{2}-\frac{1}{2}\\ \left[\left[x=-\frac{\sqrt{5}}{2}-\frac{1}{2}\&comma;-\frac{\sqrt{2\sqrt{5}+2+4a}}{2}<y\&comma;y<\frac{\sqrt{2\sqrt{5}+2+4a}}{2}\right]\&comma;\left[x=\frac{\sqrt{5}}{2}-\frac{1}{2}\&comma;-\frac{\sqrt{-2\sqrt{5}+2+4a}}{2}<y\&comma;y<\frac{\sqrt{-2\sqrt{5}+2+4a}}{2}\right]\right]& \frac{\sqrt{5}}{2}-\frac{1}{2}<a\end{array}\right.$

$\mathrm{SemiAlgebraic}\left(\left\{0<x+y^2\&comma;x^2+x-1=0\right\}\right)$

$\left[\left[x=-\frac{\sqrt{5}}{2}-\frac{1}{2}\&comma;y<-\frac{\sqrt{2+2\sqrt{5}}}{2}\right]\&comma;\left[x=-\frac{\sqrt{5}}{2}-\frac{1}{2}\&comma;\frac{\sqrt{2+2\sqrt{5}}}{2}<y\right]\&comma;\left[x=\frac{\sqrt{5}}{2}-\frac{1}{2}\&comma;y=y\right]\right]$

$\mathrm{SemiAlgebraic}\left(\left\{x^2+y^2<9\&comma;x^2+x-1<0\right\}\&comma;\left\{x\&comma;y\right\}\right)$

$\left[\left[-\frac{\sqrt{5}}{2}-\frac{1}{2}<x\&comma;x<\frac{\sqrt{5}}{2}-\frac{1}{2}\&comma;-\sqrt{-x^2+9}<y\&comma;y<\sqrt{-x^2+9}\right]\right]$

$\mathrm{SemiAlgebraic}\left(\left\{x^2+y^2<9\&comma;x^2-y<0\right\}\&comma;\left\{x\&comma;y\right\}\right)$

$\left[\left[\mathrm{RootOf}\left({\mathrm{\_Z}}^4+{\mathrm{\_Z}}^2-9\&comma;\mathrm{-1.602}..\mathrm{-1.593}\right)<x\&comma;x<\mathrm{RootOf}\left({\mathrm{\_Z}}^4+{\mathrm{\_Z}}^2-9\&comma;1.593..1.602\right)\&comma;x^2<y\&comma;y<\sqrt{-x^2+9}\right]\right]$

The SolveTools[SemiAlgebraic] command was introduced in Maple 16.

For more information on Maple 16 changes, see Updates in Maple 16.

The parameters option was updated in Maple 17.