# The Representation of Solution Sets

Any combination of equations or inequalities can be thought of as implicitly defining a region in some kind of space. The fundamental function of

Reduce is to turn this type of implicit description into an explicit one.

An implicit description in terms of equations or inequalities is sufficient if one just wants to test whether a point specified by values of variables is in the region. But to understand the structure of the region, or to generate points in it, one typically needs a more explicit description, of the kind obtained from

Reduce.

Here are inequalities that implicitly define a semicircular region.

Out[1]= | |

This shows that the point

lies in the region.

Out[2]= | |

Reduce gives a more explicit representation of the region.

Out[3]= | |

If we pick a value for

consistent with the first inequality, we then immediately get an explicit inequality for

.

Out[4]= | |

Reduce is set up to describe regions by first giving fixed conditions for

, then giving conditions for

that depend on

, then conditions for

that depend on

and

, and so on. This structure has the feature that it allows one to pick points by successively choosing values for each of the

in turn—in much the same way as when one uses iterators in functions like

Table.

This gives a representation for the region in which one first picks a value for

, then

.

Out[5]= | |

In some simple cases the region defined by a system of equations or inequalities will end up having only one component. In such cases, the output from

Reduce will be of the form

where each of the

is an equation or inequality involving variables up to

.

In most cases, however, there will be several components, represented by output containing forms such as

.

Reduce typically tries to minimize the number of components used in describing a region. But in some cases multiple parametrizations may be needed to cover a single connected component, and each one of these will appear as a separate component in the output from

Reduce.

In representing solution sets, it is common to find that several components can be described together by using forms such as

.

Reduce by default does this so as to return its results as compactly as possible. You can use

LogicalExpand to generate an expanded form in which each component appears separately.

In generating the most compact results,

Reduce sometimes ends up making conditions on later variables

depend on more of the earlier

than is strictly necessary. You can force

Reduce to generate results in which a particular

only has minimal dependence on earlier

by giving the option

True. Usually this will lead to much larger output, although sometimes it may be easier to interpret.

By default,

Reduce expresses the condition on

in terms of

.

Out[6]= | |

Backsubstituting allows conditions for

to be given without involving

.

Out[7]= | |

CylindricalDecomposition[expr,{x_{1},x_{2},...}] |

| generate the cylindrical algebraic decomposition of the region defined by expr |

GenericCylindricalDecomposition[expr,{x_{1},x_{2},...}] |

| find the full-dimensional part of the decomposition of the region defined by expr, together with any hypersurfaces containing the rest of the region |

SemialgebraicComponentInstances[expr,{x_{1},x_{2},...}] |

| give at least one point in each connected component of the region defined by expr |

Cylindrical algebraic decomposition.

For polynomial equations or inequalities over the reals, the structure of the result returned by

Reduce is typically a

*cylindrical algebraic decomposition* or

*CAD*. Sometimes

Reduce can yield a simpler form. But in all cases you can get the complete CAD by using

CylindricalDecomposition. For systems containing inequalities only,

GenericCylindricalDecomposition gives you "most" of the solution set and is often faster.

Here is the cylindrical algebraic decomposition of a region bounded by a hyperbola.

Out[8]= | |

This gives the two-dimensional part of the solution set along with a curve containing the boundary.

Out[9]= | |

This finds solutions from both parts of the solution set.

Out[10]= | |

The results include a few points from each piece of the solution set.

Out[11]= | |