The CellsWithSolutions($m,k$) calling sequence returns the indices of all cells where the system

has exactly $k$ real solutions.

The CellsWithSolutions($m,l..r$) calling sequence returns the indices of all cells where the system above has between $l$ and $r$ (inclusive) real solutions.

If there are no cells with the requested number of solutions, the empty list is returned.

This command is part of the RootFinding[Parametric] package, so it can be used in the form CellsWithSolutions(..) 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][CellsWithSolutions](..).

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

$m≔\mathrm{CellDecomposition}\left(\left[x^6+a{y}^2-b=0\,a{x}^2+y^2-bxy=0\right]\,\left[x\,y\right]\right)\:$

$m:-\mathrm{SamplePoints}$

$\left[\left[a=\mathrm{-4}\,b=\mathrm{-3}\right]\,\left[a=\mathrm{-2}\,b=\mathrm{-3}\right]\,\left[a=1\,b=\mathrm{-3}\right]\,\left[a=3\,b=\mathrm{-3}\right]\,\left[a=\mathrm{-2}\,b=\mathrm{-1}\right]\,\left[a=\mathrm{-1}\,b=\mathrm{-1}\right]\,\left[a=0\,b=\mathrm{-1}\right]\,\left[a=1\,b=\mathrm{-1}\right]\,\left[a=\mathrm{-2}\,b=1\right]\,\left[a=\mathrm{-1}\,b=1\right]\,\left[a=0\,b=1\right]\,\left[a=1\,b=1\right]\,\left[a=\mathrm{-4}\,b=3\right]\,\left[a=\mathrm{-2}\,b=3\right]\,\left[a=1\,b=3\right]\,\left[a=3\,b=3\right]\right]$

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

$\left[\left[1\,8\right]\,\left[2\,4\right]\,\left[3\,0\right]\,\left[4\,0\right]\,\left[5\,8\right]\,\left[6\,4\right]\,\left[7\,0\right]\,\left[8\,0\right]\,\left[9\,4\right]\,\left[10\,4\right]\,\left[11\,4\right]\,\left[12\,0\right]\,\left[13\,4\right]\,\left[14\,4\right]\,\left[15\,4\right]\,\left[16\,0\right]\right]$

$\mathrm{CellsWithSolutions}\left(m\,4\right)$

$\left[2\,6\,9\,10\,11\,13\,14\,15\right]$

$\mathrm{CellsWithSolutions}\left(m\,0..3\right)$

$\left[3\,4\,7\,8\,12\,16\right]$