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RegularChains[ConstructibleSetTools][RegularSystemDifference] - compute the difference of two regular systems

Calling Sequence

RegularSystemDifference(rs1, rs2, R)

Parameters

rs1, rs2

-

regular systems of R

R

-

polynomial ring

Description

• 

The command RegularSystemDifference(rs1, rs2, R) returns a constructible set which is the difference of rs1 and rs2.

• 

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

Examples

withRegularChains:

withChainTools:

withConstructibleSetTools:

Define a polynomial ring.

R:=PolynomialRingx,y,z

R:=polynomial_ring

(1)

Define a set of polynomials of R.

sys:=zx2+y+z,y2+z

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

(2)

The command Triangularize (with lazard option) decomposes the common solutions of the polynomial system sys by means of regular chains.

dec:=Triangularizesys,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 their 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,rc2:=dec1,dec2

rc1,rc2:=regular_chain,regular_chain

(5)

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

h1,h2:=x,x+z

h1,h2:=x,x+z

(6)

To obtain a regular system, first check whether 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.

rs1:=RegularSystemrc1,h1,R;rs2:=RegularSystemrc2,h2,R

rs1:=regular_system

rs2:=regular_system

(8)

The command RegularSystemDifference computes the set theoretical difference of two sets defined by regular systems. The output is a list of regular systems which forms a constructible set.

cs:=RegularSystemDifferencers1,rs2,R

cs:=constructible_set

(9)

To view the output, use the following sequence of commands.

lrs:=RepresentingRegularSystemscs,R

lrs:=regular_system

(10)

lrc:=mapRepresentingChain,lrs,R

lrc:=regular_chain

(11)

eqs:=mapEquations,lrc,R

eqs:=zx2+y+z,y2+z

(12)

ineqs:=mapRepresentingInequations,lrs,R

ineqs:=x,z

(13)

Alternatively, you could use the Info command.

Infocs,R

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

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

(14)

See Also

ConstructibleSet, ConstructibleSetTools, Difference, RegularChains, RegularSystem, RepresentingChain, RepresentingInequations


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