RegularChains[ConstructibleSetTools] - Maple Programming Help

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RegularChains[ConstructibleSetTools]

 CylindricalDecompose
 compute a cylindrical decomposition of the complex space
 SeparateZeros
 separate the zeros of a univariate parametric polynomial

 Calling Sequence CylindricalDecompose(F, R) CylindricalDecompose(F, R, 'output'='piecewise') CylindricalDecompose(F, R, 'output'='tree') SeparateZeros(rs, R)

Parameters

 R - polynomial ring F - list of polynomials of R 'output'='piecewise' - (optional) boolean flag 'output'='tree' - (optional) boolean flag rs - regular system

Description

 • 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 squarefree 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.

Examples

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

Define a ring of polynomials.

 > $R≔\mathrm{PolynomialRing}\left(\left[x,a,b\right]\right)$
 ${R}{:=}{\mathrm{polynomial_ring}}$ (1)

Define a polynomial set of R.

 > $F≔\left[a{x}^{2}-b\right]$
 ${F}{:=}\left[{a}{}{{x}}^{{2}}{-}{b}\right]$ (2)

Compute a cylindrical decomposition of the 3-D complex space.

 > $\mathrm{cd}≔\mathrm{CylindricalDecompose}\left(F,R\right)$
 ${\mathrm{cd}}{:=}{{}\begin{array}{cc}{{}\begin{array}{cc}{1}& {a}{=}{0}\\ {{}\begin{array}{cc}{1}& {x}{=}{0}\\ {1}& {\mathrm{otherwise}}\end{array}& {\mathrm{otherwise}}\end{array}& {b}{=}{0}\\ {{}\begin{array}{cc}{1}& {a}{=}{0}\\ {{}\begin{array}{cc}{1}& {{x}}^{{2}}{}{a}{-}{b}{=}{0}\\ {1}& {\mathrm{otherwise}}\end{array}& {\mathrm{otherwise}}\end{array}& {\mathrm{otherwise}}\end{array}$ (3)

Define a regular chain of R.

 > $\mathrm{rc}≔\mathrm{Chain}\left(\left[a{x}^{2}-b\right],\mathrm{Empty}\left(R\right),R\right)$
 ${\mathrm{rc}}{:=}{\mathrm{regular_chain}}$ (4)

Define a regular system of R.

 > $\mathrm{rs}≔\mathrm{RegularSystem}\left(\mathrm{rc},R\right);$$\mathrm{Info}\left(\mathrm{rs},R\right)$
 ${\mathrm{rs}}{:=}{\mathrm{regular_system}}$
 $\left[\left[{a}{}{{x}}^{{2}}{-}{b}\right]{,}\left[{1}\right]\right]$ (5)

Separate its zeros, regarding all variables except the largest as parameters.

 > $\mathrm{pairs}≔\mathrm{SeparateZeros}\left(\mathrm{rs},R\right)$
 ${\mathrm{pairs}}{:=}\left[\left[{\mathrm{constructible_set}}{,}\left[{a}{}{{x}}^{{2}}{-}{b}\right]\right]{,}\left[{\mathrm{constructible_set}}{,}\left[{x}\right]\right]\right]$ (6)

The output consists of two pairs. Display the constructible sets.

 > $\mathrm{seq}\left(\mathrm{Info}\left({\mathrm{pair}}_{1},R\right),\mathrm{pair}=\mathrm{pairs}\right)$
 $\left[\left[{}\right]{,}\left[{{a}}^{{2}}{,}{-}{b}\right]\right]{,}\left[\left[{b}\right]{,}\left[{a}\right]\right]$ (7)