The command PartialCylindricalAlgebraicDecomposition returns llr a list of points in the Euclidean space of dimension d, where d the number of variables in R.

Each point in llr is a sample point of a d dimensional connected open set, which is a cell of a Cylindrical Algebraic Decomposition (CAD) induced by the polynomial p and the polynomials in lp, under the variable projection order given by R. Recall that the variables in R are sorted in decreasing order.

If lp is not an empty list, then the points which do not satisfy q > 0 for all polynomial q in lp are discarded; otherwise, the points are in one-to-one correspondence to all the d dimensional CAD cells.

The coordinates of all these points are rational numbers, and the ith coordinate of each point of llr corresponds the ith variable of R.

The base field of R is the field of rational numbers.

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

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

$R≔\mathrm{PolynomialRing}\left(\left[x\,y\,t\right]\right)\:$

$\mathrm{PartialCylindricalAlgebraicDecomposition}\left(y\,\left[\right]\,R\right)$

$\left[\left[0\,-\frac{1}{2}\,0\right]\,\left[0\,\frac{1}{2}\,0\right]\right]$

$\mathrm{PartialCylindricalAlgebraicDecomposition}\left(y\,\left[x\right]\,R\right)$

$\left[\left[\frac{1}{2}\,-\frac{1}{2}\,0\right]\,\left[\frac{1}{2}\,\frac{1}{2}\,0\right]\right]$

$\mathrm{PartialCylindricalAlgebraicDecomposition}\left(yx\,\left[\right]\,R\right)$

$\left[\left[-\frac{1}{2}\,-\frac{1}{2}\,0\right]\,\left[\frac{1}{2}\,-\frac{1}{2}\,0\right]\,\left[-\frac{1}{2}\,\frac{1}{2}\,0\right]\,\left[\frac{1}{2}\,\frac{1}{2}\,0\right]\right]$

$\mathrm{PartialCylindricalAlgebraicDecomposition}\left(yx\,\left[x\right]\,R\right)$

$\left[\left[\frac{1}{2}\,-\frac{1}{2}\,0\right]\,\left[\frac{1}{2}\,\frac{1}{2}\,0\right]\right]$