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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

with(RegularChains): with(ChainTools):

R := PolynomialRing([x, y, z]);

Rpolynomial_ring

(1)

rc := Empty(R);

rcregular_chain

(2)

rc := Chain([z*(z-1), y*(y-2)], rc, R); Equations(rc, R);

rcregular_chain

y22y,z2z

(3)

p := z* x+y;

pzx+y

(4)

reg, sing := op(Regularize(p, rc, R));

reg,singregular_chain,regular_chain,regular_chain,regular_chain

(5)

map(Equations, reg, R);

y2,z,y,z1,y2,z1

(6)

map(Equations, sing, R);

y,z

(7)

[seq(SparsePseudoRemainder(p, reg[i], R), i=1..nops(reg))];

2,x,x+2

(8)

seq(SparsePseudoRemainder(p, sing[i], R), i=1..nops(sing));

0

(9)

See Also

Chain

Empty

Equations

Inverse

IsRegular

IsStronglyNormalized

PolynomialRing

RegularChains

RegularizeDim0

RegularizeInitial

SparsePseudoRemainder