DefiningSet - Maple Help
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RegularChains[ParametricSystemTools]

  

DefiningSet

  

compute the defining set of a regular chain

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

DefiningSet(rc, d, R)

Parameters

rc

-

regular chain

d

-

number of parameters

R

-

polynomial ring

Description

• 

The command DefiningSet(rc, d, R) returns the defining set of rc with respect to the last d variables, regarded as parameters. This is a constructible set .

• 

Given a positive integer d, the regular chain rc can be split into two parts. Denote by  the set of the polynomials in rc involving only the last d variables, and denote by  the other polynomials of rc. Certainly, both  and  are regular chains.

• 

Let  be the quasi-component of . For a point  in , after specializing  at , two situations arise:

  

(1) either  is not a regular chain anymore;

  

(2) or  is still a regular chain.

  

There is a subtle point: after specializing  at , it might happen that it is still a regular chain, but its shape changes. In other words, the degree of the geometric object given by  could change. The term specialize well, defined below, takes these cases into account.

• 

The regular chain rc specializes well at a point  of  if  is a regular chain after specialization and no initial of polynomials in rc1 vanish during the specialization.

• 

The defining set of rc with respect to the last d variables consists of the points in  at which rc specializes well.

• 

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

Examples

(1)

Consider the following parametric polynomial system F.

(2)

For different values of u and v, the solution set has a different nature. For example, u=0 and v=0 is a degenerate case: x=0 and y can be any value. To understand more about F, first decompose F into a set of regular chains.

(3)

(4)

The first regular chain is simple. For all values of u and v, it is well-specialized.

(5)

For the last one, its defining set is given by  and ,  and the inequality is to ensure that rc1 specializes well.

(6)

See Also

ComprehensiveTriangularize

ConstructibleSet

DiscriminantSet

Info

ParametricSystemTools

PreComprehensiveTriangularize

RegularChains

Triangularize

 


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