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RegularChains[ConstructibleSetTools]

  

RepresentingRegularSystems

  

return the list of regular systems in a constructible set

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

RepresentingRegularSystems(cs, R)

Parameters

cs

-

constructible set

R

-

polynomial ring

Description

• 

The command RepresentingRegularSystems(cs,R) returns a list of regular systems which defines the constructible set cs, that is, a list of regular systems (whose polynomials belong to R) such that the union of their zero sets is exactly equal to cs.

• 

Recall that every constructible set built by the ConstructibleSetTools module is in fact represented by a list of regular systems representing it in the above sense.

• 

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.

• 

The command RepresentingRegularSystems is part of the RegularChains[ConstructibleSetTools] package, so it can be used in the form RepresentingRegularSystems(..) 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][RepresentingRegularSystems](..).

Examples

withRegularChains:

withConstructibleSetTools:

First, define a polynomial ring R and two polynomials of R.

RPolynomialRingx,y,u,v

R:=polynomial_ring

(1)

fux+v;gvy+u

f:=ux+v

g:=vy+u

(2)

Using GeneralConstruct, construct a constructible set from the common solutions of f and g which do not cancel u2+v21 

csGeneralConstructf,g,u2+v21,R

cs:=constructible_set

(3)

Now retrieve the regular systems from cs.

lrsRepresentingRegularSystemscs,R

lrs:=regular_system,regular_system

(4)

Next extract the representing chains and inequations

lrcmapRepresentingChain,lrs,R

lrc:=regular_chain,regular_chain

(5)

mapEquations,lrc,R

ux+v,vy+u,u,v

(6)

mapRepresentingInequations,lrs,R

u2+v21,

(7)

The first inequation is u2+v21 since this polynomial can vanish inside the quasi-component of the first regular chain.

The second inequation is simply 1 since u2+v21 cannot vanish inside the quasi-component of the second regular chain.

See Also

ConstructibleSet

ConstructibleSetTools

GeneralConstruct

Info

QuasiComponent

RegularChains

RegularSystem

RepresentingChain

 


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