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RegularChains[ConstructibleSetTools]

  

ConstructibleSet

  

construct a constructible set from a list or set of regular systems

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

ConstructibleSet(lrs, R)

Parameters

lrs

-

list or set of regular systems

R

-

polynomial ring

Description

• 

The command ConstructibleSet(lrs, R) returns a constructible set defined by the list lrs of regular systems.

• 

A point belongs to a constructible set if and only if it is a solution of one of its defining regular systems. That is, a constructible set is the union of the solution sets of its defining regular systems.

• 

Since a regular system always defines a nonempty set, a constructible set is empty if and only if its list of defining regular systems is empty.

• 

This command is part of the RegularChains[ConstructibleSetTools] package, so it can be used in the form ConstructibleSet(..) only after executing the command with(RegularChains[ConstructibleSetTools]).  However, it can always be accessed through the long form of the command by using RegularChains[ConstructibleSetTools][ConstructibleSet](..).

• 

See ConstructibleSetTools and RegularChains for the related mathematical concepts, in particular for the ideas of a constructible set, a regular system, and a regular chain.

Examples

This example demonstrates how to build a constructible set structure.

withRegularChains:

withConstructibleSetTools:

First, define a polynomial ring.

RPolynomialRingx,y,a,b,c,d

R:=polynomial_ring

(1)

Consider the following linear polynomial system.

sysax+by,cx+dy

sys:=ax+by,cx+dy

(2)

The command Triangularize with lazard option decomposes the solution set by means of regular chains. Each regular chain describes a group of solutions with certain mathematical meaning. See RegularChains for more information.

decTriangularizesys,R,output=lazard

dec:=regular_chain,regular_chain,regular_chain,regular_chain,regular_chain,regular_chain

(3)

To build constructible sets, you first need to create regular systems. For simplicity, just let 1 be the inequation part of each regular system.

lrsmapRegularSystem,dec,1,R

lrs:=regular_system,regular_system,regular_system,regular_system,regular_system,regular_system

(4)

Then lrs is a list of regular systems by which you can create a constructible set cs.

csConstructibleSetlrs,R

cs:=constructible_set

(5)

Use Info to see its internal defining polynomials.

Infocs,R

x,y,1,cx+yd,dabc,1,y,a,c,1,x,b,d,1,ax+yb,c,d,1,a,b,c,d,1

(6)

See Also

ConstructibleSetTools

Info

QuasiComponent

RegularChains

RegularSystem

RepresentingChain

RepresentingInequations

RepresentingRegularSystems

Triangularize

 


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