CylindricalDecompose(F, R) returns an F-invariant cylindrical decomposition of the n-dimensional complex space, where n is the number of variables in R.

A cylindrical decomposition of the n-dimensional complex space is a partition of the whole space into cells that are constructible sets, such that the cells in the partition are cylindrically arranged, that is, the projection of any two cells onto any lower dimensional complex space are either equal or disjoint. This decomposition is called F-invariant if for any given cell, either all polynomials of F are identically zero in that cell, or none of them admits a zero in that cell.

The output is represented by a tree or a piecewise function. The default is the piecewise format.

SeparateZeros(rs, R) returns a list, say pairs. Each item of pairs is a pair [cs, polys] where cs is a constructible set and polys is a list of nonconstant polynomials such that for each point u of cs, the initial of each polynomial p in polys does not vanish at u and all the polynomials p(u) are square free and pairwise coprime. Moreover, above each cs, the zero set of rs is equal to the union of the zero sets of the polynomials in polys.

Roughly speaking, SeparateZeros(rs, R) decomposes the zeros of rs regarding all variables but the largest one as parameters. Since the input is a regular system, one polynomial in rs has a non-zero degree in this largest variable; the other polynomials in rs define constraints on the parameters.

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

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

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

$R≔\mathrm{PolynomialRing}\left(\left[x\,a\,b\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$F≔\left[a{x}^2-b\right]$

$F≔\left[a{x}^2-b\right]$

$\mathrm{cd}≔\mathrm{CylindricalDecompose}\left(F\,R\right)$

$\mathrm{cd}≔\mathrm{c\_c\_d}$

$\mathrm{rc}≔\mathrm{Chain}\left(\left[a{x}^2-b\right]\,\mathrm{Empty}\left(R\right)\,R\right)$

$\mathrm{rc}≔\mathrm{regular\_chain}$

$\mathrm{rs}≔\mathrm{RegularSystem}\left(\mathrm{rc}\,R\right)\;$$\mathrm{Info}\left(\mathrm{rs}\,R\right)$

$\mathrm{rs}≔\mathrm{regular\_system}$

$\left[\left[a{x}^2-b\right]\,\left[1\right]\right]$

$\mathrm{pairs}≔\mathrm{SeparateZeros}\left(\mathrm{rs}\,R\right)$

$\mathrm{pairs}≔\left[\left[\mathrm{constructible\_set}\,\left[a{x}^2-b\right]\right]\,\left[\mathrm{constructible\_set}\,\left[x\right]\right]\right]$

$\mathrm{seq}\left(\mathrm{Info}\left(\mathrm{pair}\left[1\right]\,R\right)\,\mathrm{pair}=\mathrm{pairs}\right)$

$\left[\left[\right]\,\left[a^2\,-b\right]\right],\left[\left[b\right]\,\left[a\right]\right]$