Built-in Mathematica Symbol

CylindricalDecomposition

CylindricalDecomposition[ineqs, {x₁, x₂, ...}]
finds a decomposition of the region represented by the inequalities ineqs into cylindrical parts whose directions correspond to the successive

.

MORE INFORMATION

CylindricalDecomposition assumes that all variables are real.

Lists or logical combinations of inequalities can be given.

CylindricalDecomposition returns inequalities whose bounds in general involve algebraic functions.

EXAMPLES

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Basic Examples (1)

Find a cylindrical decomposition of the unit disc:

In[1]:=

Out[1]=

Scope (5)

For univariate polynomials the result consists of intervals:

In general individual points can occur:

This is the form for any logical combination as well:

For multivariate polynomials the result is in cylinder form

:

In general several cylinders will result:

Plot the individual cylinders using RegionPlot:

By changing the order of variables, the cylinders take the form

:

Plot the individual cylinders:

Here cylinders of dimensions

,

, and

occur in the result:

Three- and four-dimensional decompositions:

Generalizations & Extensions (4)

CylindricalDecomposition also allows quantified formulas:

Coefficients can include real algebraic numbers:

Coefficients can include real exact transcendental numbers:

Functions can be real algebraic:

Options (1)

This computation takes a long time due to high degrees of algebraic numbers involved:

This finds a decomposition using WorkingPrecision->100, but the result may be incorrect:

Properties & Relations (8)

Use RegionPlot to visualize 2D semialgebraic sets:

Use RegionPlot3D to visualize 3D semialgebraic sets:

Resolve performs quantifier elimination, and may avoid computing cylindrical decomposition:

Reduce in addition deals with different domains and transcendental functions:

Use FindInstance to find points that satisfy equations and inequalities:

SemialgebraicComponentInstances will give sample points in each cylinder:

CylindricalDecomposition merges several cylinders to get a more compact representation:

GenericCylindricalDecomposition will compute the full dimensional part only:

The output and input are equal as sets:

Points are simultaneously inside or outside of the sets:

Possible Issues (2)

CylindricalDecomposition requires exact, infinite-precision input:

Rationalize will convert inexact numbers to exact ones:

In general the output can be in a nested and more compact form:

Flatten the result into disjunctive normal form without splitting the inequalities:

Neat Examples (1)

Semi-algebraic sets are quite general: