The CellDescription command returns a list of lists of the form $\left[p\,i\,a\,q\,j\right]$ where

a is a parameter

p and q are polynomials in the parameters

i and j are non-negative integers

The CellDescription($m,k$) calling sequence returns a description of the $k$th cell in $m$ in terms of real roots of some projection polynomials.

The solution record m must have been computed with the option output=cad or without using the output keyword.

Each inner list, $\left[p\,i\,a\,q\,j\right]$, in the result is to be interpreted as follows: the $a$-coordinate of a point $u$ lying in the interior of the $k$th cell is greater than the $i$th real root of the polynomial $p$ and less than the $j$th real root of the polynomial $q$.

If an inner list is of the form $\left[p\,i\,a\,\mathrm{\infty }\,0\right]$, then this means that the $a$-coordinate is unbounded from above, and similarly, if an inner list is of the form  $\left[-\mathrm{\infty }\,0\,a\,q\,j\right]$, then the $a$-coordinate is unbounded from below.

The polynomials $p$ and $q$ in each inner list contain only the parameters from the current and all earlier lists. So the polynomials in the first inner list are univariate, the ones in the second inner list are bivariate, etc.

The result, [[$p_1,i_1,a_1,q_1,j_1$], [$p_2,i_2,a_2,q_2,j_2$], ...], can be used to sample a cell as follows: Compute the $i_1$th real root of the univariate polynomial $p_1$ and the $j_1$th real root of the univariate polynomial $p_2$ (for example, using RootFinding[Isolate]), and pick a value for the $a_1$-coordinate between those two roots. Then substitute that value of $a_1$ into $p_2$ and $q_2$, turning these into univariate polynomials in $a_2$. In the same way as above, compute their $i_2$th and $j_2$th roots, respectively, and pick a value for the $a_2$-coordinate in between those two roots. Continue in a similar fashion.

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

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

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

$m≔\left[\begin{array}{lll}\mathrm{Equations}& \=& \left[a{y}^2+x^2-b\,a{x}^2+by\right]\\ \mathrm{Inequalities}& \=& \left[\right]\\ \mathrm{Filter}& \=& 0\ne 1\\ \mathrm{Variables}& \=& \left[x\,y\right]\\ \mathrm{Parameters}& \=& \left[a\,b\right]\\ \mathrm{DiscriminantVariety}& \=& \left[\left[a\right]\,\left[b\right]\,\left[4a^3+b\right]\right]\\ \mathrm{ProjectionPolynomials}& \=& \left[\left[b\right]\,\left[a\,4a^3+b\right]\right]\\ \mathrm{SamplePoints}& \=& \left[\left[a=\mathrm{-1}\,b=\mathrm{-1}\right]\,\left[a=\frac{95196942993367727481803}{302231454903657293676544}\,b=\mathrm{-1}\right]\,\left[a=1\,b=\mathrm{-1}\right]\,\left[a=\mathrm{-1}\,b=1\right]\,\left[a=-\frac{177318124088197}{562949953421312}\,b=1\right]\,\left[a=1\,b=1\right]\right]\end{array}\right.$

$\mathrm{CellDescription}\left(m\,5\right)$

$\left[\left[b\,1\,b\,\mathrm{\infty }\,0\right]\,\left[4a^3+b\,1\,a\,a\,1\right]\right]$

$u_2$ is greater than the $1$st (and only) real root of $b=0$, that is, $0<u_2$; and

$u_1$ is greater than the first (and only) real root of $4a^3+u_2=0$ and less than the first (and only) real root of $a=0$, that is,

$\mathrm{CellDescription}\left(m\&comma;4\right)$

$\left[\left[b\&comma;1\&comma;b\&comma;\mathrm{\infty }\&comma;0\right]\&comma;\left[-\mathrm{\infty }\&comma;0\&comma;a\&comma;4a^3+b\&comma;1\right]\right]$

$\mathrm{CellPlot}\left(m\&comma;\left[4\&comma;5\right]\&comma;'\mathrm{samplepoints}'\right)$