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RegularChains[ChainTools]

  

Regularize

  

make a polynomial regular or null with respect to a regular chain

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

Regularize(p, rc, R)

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

Regularize(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 Regularize(p, rc, R) returns a list made of two lists. The first one consists of regular chains reg_i such that p is regular modulo the saturated ideal of reg_i. The second one consists of regular chains sing_i such that p is null modulo the saturated ideal of sing_i.

• 

In addition, the union of the regular chains of these lists is a decomposition of rc in the sense of Kalkbrener.

• 

If 'normalized'='yes' is passed, all the returned regular chains are normalized.

• 

If 'normalized'='strongly' is passed, all the returned regular chains are strongly normalized.

• 

If 'normalized'='yes' is present, rc must be normalized.

• 

If 'normalized'='strongly' is present, rc must be strongly normalized.

• 

The command RegularizeDim0 implements another algorithm with the same purpose as that of the command Regularize. However it is specialized to zero-dimensional regular chains in prime characteristic. When both algorithms apply, the latter usually outperforms the former one.

• 

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

Examples

withRegularChains:withChainTools:

RPolynomialRingx,y,z

R:=polynomial_ring

(1)

rcEmptyR

rc:=regular_chain

(2)

rcChainzz1,yy2,rc,R;Equationsrc,R

rc:=regular_chain

y22y,z2z

(3)

pzx+y

p:=xz+y

(4)

reg,singopRegularizep,rc,R

reg,sing:=regular_chain,regular_chain,regular_chain,regular_chain

(5)

mapEquations,reg,R

y2,z,y,z1,y2,z1

(6)

mapEquations,sing,R

y,z

(7)

seqSparsePseudoRemainderp,regi,R,i=1..nopsreg

2,x,2+x

(8)

seqSparsePseudoRemainderp,singi,R,i=1..nopssing

0

(9)

See Also

Chain

Empty

Equations

Inverse

IsRegular

IsStronglyNormalized

PolynomialRing

RegularChains

RegularizeDim0

RegularizeInitial

SparsePseudoRemainder

 


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