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RegularChains[ChainTools]

  

Construct

  

constructs regular chains

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

Construct(p, rc, R)

Construct(p, rc, R, 'normalized'='yes')

Construct(p, rc, R, 'normalized'='strongly')

Parameters

p

-

polynomial of R

rc

-

regular chain of R

R

-

polynomial ring

'normalized'='yes'

-

(optional) boolean flag

'normalized'='strongly'

-

(optional) boolean flag

Description

• 

The command Construct(p, rc, R) returns a list of regular chains rci which form a triangular decomposition of the regular chain obtained by extending rc with p.

• 

This assumes that p is a non-constant with main variable greater than any algebraic variable of rc, and that the initial of p is regular modulo the saturated ideal of rc. Hence p and rc form together a regular chain.

• 

Although rc with p is assumed to form a regular chain, several regular chains may be returned; this is because the polynomial p may be factorized with respect to rc in order to simplify the expressions in the regular chains rci.

• 

Such factorizations will happen if they can be performed quickly. For instance, if p involves only one variable.

• 

To avoid these possible factorizations, use RegularChains[ChainTools][Chain]

• 

If 'normalized'='yes' is present, then rc must be normalized. In addition, every returned regular chain is normalized.

• 

If 'normalized'='strongly' is present, then rc must be strongly normalized. In addition, every returned regular chain is strongly normalized.

• 

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

Examples

withRegularChains:withChainTools:

RPolynomialRingt,x,y,z

R:=polynomial_ring

(1)

pzz2+2z+1

pz:=z2+2z+1

(2)

pyy2+z

py:=y2+z

(3)

ptt3+yz

pt:=t3+yz

(4)

rcEmptyR

rc:=regular_chain

(5)

rc1Constructpz,rc,R

rc1:=regular_chain

(6)

rc1rc11;Equationsrc1,R

rc1:=regular_chain

z+1

(7)

rc2Constructpy,rc1,R

rc2:=regular_chain,regular_chain

(8)

rc2rc21;Equationsrc2,R

rc2:=regular_chain

y1,z+1

(9)

rc3Constructpt,rc2,R

rc3:=regular_chain,regular_chain

(10)

mapEquations,rc3,R

t1,y1,z+1,t2+t+1,y1,z+1

(11)

See Also

Chain

ChainTools

Empty

Equations

ListConstruct

PolynomialRing

RegularChains

 


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