 SemiAlgebraic System Solving - Maple Help

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SemiAlgebraic System Solving

The SolveTools[SemiAlgebraic] command has been integrated directly into the solve command, such that many systems involving non-linear polynomial inequalities that could not be solved previously, are solved.

In Maple 16, no solutions were found for the following system, but in Maple 17 it is easily solved.

 > $\mathrm{sol}:=\mathrm{solve}\left(\left\{0\le x-{y}^{2},{x}^{2}+{y}^{2}<9,0<-y+3x-2\right\},\left[x,y\right]\right)$
 ${\mathrm{sol}}{:=}\left[\left[{x}{=}{1}{,}{y}{<}{1}{,}{-}{1}{\le }{y}\right]{,}\left[{x}{=}{-}\frac{{1}}{{2}}{+}\frac{{1}}{{2}}{}\sqrt{{37}}{,}{y}{<}\frac{{1}}{{2}}{}\sqrt{{-}{2}{+}{2}{}\sqrt{{37}}}{,}{-}\frac{{1}}{{2}}{}\sqrt{{-}{2}{+}{2}{}\sqrt{{37}}}{<}{y}\right]{,}\left[{x}{=}\frac{{3}}{{5}}{+}\frac{{1}}{{10}}{}\sqrt{{86}}{,}{y}{\le }\frac{{1}}{{10}}{}\sqrt{{60}{+}{10}{}\sqrt{{86}}}{,}{-}\frac{{1}}{{10}}{}\sqrt{{60}{+}{10}{}\sqrt{{86}}}{\le }{y}\right]{,}\left[{x}{<}\frac{{3}}{{5}}{+}\frac{{1}}{{10}}{}\sqrt{{86}}{,}{1}{<}{x}{,}{y}{=}\sqrt{{x}}\right]{,}\left[{x}{<}\frac{{3}}{{5}}{+}\frac{{1}}{{10}}{}\sqrt{{86}}{,}{1}{<}{x}{,}{y}{=}{-}\sqrt{{x}}\right]{,}\left[{x}{<}{1}{,}\frac{{4}}{{9}}{<}{x}{,}{y}{=}{-}\sqrt{{x}}\right]{,}\left[{x}{<}{-}\frac{{1}}{{2}}{+}\frac{{1}}{{2}}{}\sqrt{{37}}{,}\frac{{3}}{{5}}{+}\frac{{1}}{{10}}{}\sqrt{{86}}{<}{x}{,}{y}{=}\sqrt{{x}}\right]{,}\left[{x}{<}{-}\frac{{1}}{{2}}{+}\frac{{1}}{{2}}{}\sqrt{{37}}{,}\frac{{3}}{{5}}{+}\frac{{1}}{{10}}{}\sqrt{{86}}{<}{x}{,}{y}{=}{-}\sqrt{{x}}\right]{,}\left[{x}{<}\frac{{3}}{{5}}{+}\frac{{1}}{{10}}{}\sqrt{{86}}{,}{1}{<}{x}{,}{y}{<}\sqrt{{x}}{,}{-}\sqrt{{x}}{<}{y}\right]{,}\left[{x}{<}{1}{,}\frac{{4}}{{9}}{<}{x}{,}{y}{<}{3}{}{x}{-}{2}{,}{-}\sqrt{{x}}{<}{y}\right]{,}\left[{x}{<}{3}{,}{-}\frac{{1}}{{2}}{+}\frac{{1}}{{2}}{}\sqrt{{37}}{<}{x}{,}{y}{<}\sqrt{{-}{{x}}^{{2}}{+}{9}}{,}{-}\sqrt{{-}{{x}}^{{2}}{+}{9}}{<}{y}\right]{,}\left[{x}{<}{-}\frac{{1}}{{2}}{+}\frac{{1}}{{2}}{}\sqrt{{37}}{,}\frac{{3}}{{5}}{+}\frac{{1}}{{10}}{}\sqrt{{86}}{<}{x}{,}{y}{<}\sqrt{{x}}{,}{-}\sqrt{{x}}{<}{y}\right]\right]$ (1)

A solution to such a system of inequalities is a decomposition of the feasible region of the inequalities into bands as can be seen in the inequality plot:

 > $\mathrm{plots}[\mathrm{inequal}]\left(\mathrm{sol},x=0..3,y=-2..2\right)$ Additionally, the SemiAlgebraic command now can build case discussions for systems with real-valued parameters.

 > $\mathrm{SolveTools}[\mathrm{SemiAlgebraic}]\left(\left\{a{x}^{2}
 ${{}\begin{array}{cc}\left[\left[{0}{<}{x}\right]\right]& {\mathrm{And}}{}\left({a}{=}{0}{,}{0}{<}{b}\right)\\ \left[\left[{a}{<}{x}{,}{x}{<}\frac{\sqrt{{a}{}{b}}}{{a}}\right]\right]& {\mathrm{And}}{}\left({0}{<}{a}{,}{{a}}^{{3}}{<}{b}\right)\\ \left[\left[{0}{<}{x}\right]{,}\left[{a}{<}{x}{,}{x}{<}{0}\right]\right]& {\mathrm{And}}{}\left({a}{<}{0}{,}{b}{=}{0}\right)\\ \left[\left[{-}{a}{<}{x}\right]\right]& {\mathrm{And}}{}\left({a}{<}{0}{,}{b}{=}{{a}}^{{3}}\right)\\ \left[\left[{a}{<}{x}\right]\right]& {\mathrm{And}}{}\left({a}{<}{0}{,}{0}{<}{b}\right)\\ \left[\left[{-}\frac{\sqrt{{a}{}{b}}}{{a}}{<}{x}\right]\right]& {\mathrm{And}}{}\left({a}{<}{0}{,}{b}{<}{{a}}^{{3}}\right)\\ \left[\left[{-}\frac{\sqrt{{a}{}{b}}}{{a}}{<}{x}\right]{,}\left[{a}{<}{x}{,}{x}{<}\frac{\sqrt{{a}{}{b}}}{{a}}\right]\right]& {\mathrm{And}}{}\left({a}{<}{0}{,}{{a}}^{{3}}{<}{b}{,}{b}{<}{0}\right)\\ \left[{}\right]& {\mathrm{otherwise}}\end{array}$ (2) SeeAlso