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RegularChains[ConstructibleSetTools]

  

RegularSystem

  

construct a regular system from a regular chain and a list of inequations

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

RegularSystem(rc, H, R)

RegularSystem(rc, R)

RegularSystem(H, R)

RegularSystem(R)

Parameters

rc

-

regular chain

H

-

list of polynomials of R

R

-

polynomial ring

Description

• 

The command RegularSystem(rc, H, R) constructs a regular system from a regular chain and a list of inequations. Denote by WT the quasi-component of rc. Then the constructed regular system encodes those points in WT that do not cancel any polynomial in H.

• 

Each polynomial in H must be regular with respect to the regular chain rc; otherwise an error is reported.

• 

If rc is not specified, then rc is set to the empty regular chain.

• 

If H is not specified, then H is set to 1.

• 

The command RegularSystem(R) constructs the regular system corresponding to the whole space.

• 

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

• 

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

withRegularChains:

withChainTools:

withConstructibleSetTools:

Define a polynomial ring.

RPolynomialRingx,y,z

R:=polynomial_ring

(1)

Define a set of polynomials of R.

syszx2+y+z,y2+z

sys:=x2z+y+z,y2+z

(2)

decTriangularizesys,R,output=lazard

dec:=regular_chain,regular_chain

(3)

There are two groups of solutions, each of which is given by a regular chain. To view the equations, use the Equations command.

mapEquations,dec,R

zx2+y+z,y2+z,y,z

(4)

Let rc1 be the first regular chain, and rc2 be the second one.

rc1,rc2dec1,dec2

rc1,rc2:=regular_chain,regular_chain

(5)

Consider two polynomials h1 and h2; regard them as inequations.

h1,h2x,x+z

h1,h2:=x,x+z

(6)

To obtain regular systems, first check if h1 is regular with respect to rc1, and h2 is regular with respect to rc2.

IsRegularh1,rc1,R;IsRegularh2,rc2,R

true

true

(7)

Both of them are regular, thus you can build the following regular systems.

rs1RegularSystemrc1,h1,R;rs2RegularSystemrc2,h2,R

rs1:=regular_system

rs2:=regular_system

(8)

You can simply call RegularSystem(R) to build the regular system which encodes all points.

wsConstructibleSetRegularSystemR,R

ws:=constructible_set

(9)

The complement of ws must be empty.

IsEmptyComplementws,R,R

true

(10)

See Also

ConstructibleSet

ConstructibleSetTools

QuasiComponent

RegularChains

RegularSystemDifference

RepresentingChain

RepresentingInequations

RepresentingRegularSystems

 


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