 PartialCylindricalAlgebraicDecomposition - Maple Help

RegularChains[SemiAlgebraicSetTools]

 PartialCylindricalAlgebraicDecomposition
 compute a partial cylindrical algebraic decomposition Calling Sequence PartialCylindricalAlgebraicDecomposition(p, lp, R) Parameters

 R - polynomial ring p - polynomial of R lp - list of polynomials of R Description

 • 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. Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$
 > $\mathrm{with}\left(\mathrm{SemiAlgebraicSetTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[x,y,t\right]\right):$
 > $\mathrm{PartialCylindricalAlgebraicDecomposition}\left(y,\left[\right],R\right)$
 $\left[\left[{0}{,}{-}\frac{{1}}{{2}}{,}{0}\right]{,}\left[{0}{,}\frac{{1}}{{2}}{,}{0}\right]\right]$ (1)
 > $\mathrm{PartialCylindricalAlgebraicDecomposition}\left(y,\left[x\right],R\right)$
 $\left[\left[\frac{{1}}{{2}}{,}{-}\frac{{1}}{{2}}{,}{0}\right]{,}\left[\frac{{1}}{{2}}{,}\frac{{1}}{{2}}{,}{0}\right]\right]$ (2)
 > $\mathrm{PartialCylindricalAlgebraicDecomposition}\left(yx,\left[\right],R\right)$
 $\left[\left[{-}\frac{{1}}{{2}}{,}{-}\frac{{1}}{{2}}{,}{0}\right]{,}\left[\frac{{1}}{{2}}{,}{-}\frac{{1}}{{2}}{,}{0}\right]{,}\left[{-}\frac{{1}}{{2}}{,}\frac{{1}}{{2}}{,}{0}\right]{,}\left[\frac{{1}}{{2}}{,}\frac{{1}}{{2}}{,}{0}\right]\right]$ (3)
 > $\mathrm{PartialCylindricalAlgebraicDecomposition}\left(yx,\left[x\right],R\right)$
 $\left[\left[\frac{{1}}{{2}}{,}{-}\frac{{1}}{{2}}{,}{0}\right]{,}\left[\frac{{1}}{{2}}{,}\frac{{1}}{{2}}{,}{0}\right]\right]$ (4)