The $\mathrm{NumberOfSolutions}\left(m\right)$ calling sequence computes the number of real solutions of the system

and returns it as a list of lists of the form $\left[i\,n_i\right]$, where $i$ is the index of the open cell and $n_i$ is the number of real solutions of the non-parametric system resulting from substituting parameter values from this cell.

If l is given, only those open cells whose sample points lie inside the box specified by l are considered.

This command is part of the RootFinding[Parametric] package, so it can be used in the form NumberOfSolutions(..) only after executing the command with(RootFinding[Parametric]). However, it can always be accessed through the long form of the command by using RootFinding[Parametric][NumberOfSolutions](..).

$\mathrm{with}\left(\mathrm{RootFinding}\left[\mathrm{Parametric}\right]\right)\:$

$\mathrm{sys}≔\left[x^2+y^2=a\&comma;x-y=b\&comma;0<a\right]$

$\mathrm{sys}≔\left[x^2+y^2=a\&comma;x-y=b\&comma;0<a\right]$

$m≔\mathrm{CellDecomposition}\left(\mathrm{sys}\&comma;\left[x\&comma;y\right]\right)$

$m≔\left[\begin{array}{lll}\mathrm{Equations}& \&equals;& \left[x^2+y^2-a\&comma;x-y-b\right]\\ \mathrm{Inequalities}& \&equals;& \left[a\right]\\ \mathrm{Filter}& \&equals;& 0\ne 1\\ \mathrm{Variables}& \&equals;& \left[x\&comma;y\right]\\ \mathrm{Parameters}& \&equals;& \left[a\&comma;b\right]\\ \mathrm{DiscriminantVariety}& \&equals;& \left[\left[a\right]\&comma;\left[-b^2+2a\right]\right]\\ \mathrm{ProjectionPolynomials}& \&equals;& \left[\left[b\right]\&comma;\left[a\&comma;-b^2+2a\right]\right]\\ \mathrm{SamplePoints}& \&equals;& \left[\left[a=\frac{302231454903657293676531}{1208925819614629174706176}\&comma;b=\mathrm{-1}\right]\&comma;\left[a=1\&comma;b=\mathrm{-1}\right]\&comma;\left[a=\frac{302231454903657293676531}{1208925819614629174706176}\&comma;b=1\right]\&comma;\left[a=1\&comma;b=1\right]\right]\end{array}\right.$

$\mathrm{CellPlot}\left(m\&comma;'\mathrm{samplepoints}'\right)$

$\mathrm{NumberOfSolutions}\left(m\right)$

$\left[\left[1\&comma;0\right]\&comma;\left[2\&comma;2\right]\&comma;\left[3\&comma;0\right]\&comma;\left[4\&comma;2\right]\right]$

$\mathrm{NumberOfSolutions}\left(m\&comma;\left[b=0..1\right]\right)$

$\left[\left[3\&comma;0\right]\&comma;\left[4\&comma;2\right]\right]$

$\mathrm{NumberOfSolutions}\left(m\&comma;\left[a=0..\frac{1}{2}\&comma;b=0..1\right]\right)$

$\left[\left[3\&comma;0\right]\right]$

$\mathrm{NumberOfSolutions}\left(m\&comma;\left[a=0..\frac{1}{2}\&comma;b=2..3\right]\right)$

$m:-\mathrm{DiscriminantVariety}$

$\left[\left[a\right]\&comma;\left[-b^2+2a\right]\right]$

$\mathrm{eqs}≔\left[\mathrm{op}\left(m:-\mathrm{Equations}\right)\&comma;\mathrm{op}\left(\left[2\&comma;1\right]\&comma;\right)\right]$

$\mathrm{eqs}≔\left[x^2+y^2-a\&comma;x-y-b\&comma;-b^2+2a\right]$

$\mathrm{CellDecomposition}\left(\mathrm{eqs}\&comma;m:-\mathrm{Inequalities}\&comma;\left[x\&comma;y\&comma;b\right]\right)$

Error, (in RootFinding:-Parametric:-CellDecomposition) cannot solve the system: either there are infinitely many complex solutions, or there are solutions of multiplicity > 1, for almost all parameter values

$\mathrm{with}\left(\mathrm{PolynomialIdeals}\&comma;\mathrm{PolynomialIdeal}\&comma;\mathrm{Generators}\&comma;\mathrm{Simplify}\&comma;\mathrm{Radical}\right)\&colon;$

$J≔\mathrm{PolynomialIdeal}\left(\mathrm{eqs}\&comma;\mathrm{variables}=\left[x\&comma;y\&comma;b\right]\right)\&colon;$

$R≔\mathrm{Generators}\left(\mathrm{Simplify}\left(\mathrm{Radical}\left(J\right)\right)\right)$

$R≔\left\{2x-b\&comma;2y+b\&comma;b^2-2a\right\}$

$\mathrm{m2}≔\mathrm{CellDecomposition}\left(\left[\mathrm{op}\left(R\right)\right]\&comma;m:-\mathrm{Inequalities}\&comma;\left[x\&comma;y\&comma;b\right]\right)$

$\mathrm{m2}≔\left[\begin{array}{lll}\mathrm{Equations}& \&equals;& \left[2x-b\&comma;2y+b\&comma;b^2-2a\right]\\ \mathrm{Inequalities}& \&equals;& \left[a\right]\\ \mathrm{Filter}& \&equals;& 0\ne 1\\ \mathrm{Variables}& \&equals;& \left[x\&comma;y\&comma;b\right]\\ \mathrm{Parameters}& \&equals;& \left[a\right]\\ \mathrm{DiscriminantVariety}& \&equals;& \left[\left[a\right]\right]\\ \mathrm{ProjectionPolynomials}& \&equals;& \left[\left[a\right]\right]\\ \mathrm{SamplePoints}& \&equals;& \left[\left[a=1\right]\right]\end{array}\right.$

$\mathrm{NumberOfSolutions}\left(\mathrm{m2}\right)$

$\left[\left[1\&comma;2\right]\right]$

$\mathrm{SampleSolutions}\left(\mathrm{m2}\&comma;\left[a=1\right]\right)$

$\left[\left[x=\mathrm{-0.7071067812}\&comma;y=0.7071067812\&comma;b=\mathrm{-1.414213562}\right]\&comma;\left[x=0.7071067812\&comma;y=\mathrm{-0.7071067812}\&comma;b=1.414213562\right]\right]$