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RegularChains[ChainTools]

  

IsInRadical

  

test membership to the radical of a saturated ideal

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

IsInRadical(p, rc, R)

Parameters

p

-

polynomial of R

rc

-

regular chain of R

R

-

polynomial ring

Description

• 

The command IsInRadical(p, rc, R) returns true if and only if p belongs to the radical of the saturated ideal of rc.

• 

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

Examples

withRegularChains:

withChainTools:

RPolynomialRingy,x

R:=polynomial_ring

(1)

sysx2+1,y+2x2

sys:=y+2x2,x2+1

(2)

Note that this input system is already a regular chain.

outTriangularizesys,R;rcout1

out:=regular_chain

rc:=regular_chain

(3)

Equationsrc,R

y2+4xy4,x2+1

(4)

NumberOfSolutionsrc,R

4

(5)

Is y+2x in the saturated ideal of rc?

IsInSaturatey+2x,rc,R

false

(6)

Is y+2x is the radical of the saturated ideal of rc?

IsInRadicaly+2x,rc,R

true

(7)

The function Triangularize can remove the squares as follows.

outTriangularizesys,R,radical=yes;sfrcout1

out:=regular_chain

sfrc:=regular_chain

(8)

Equationssfrc,R;NumberOfSolutionssfrc,R

y+2x,x2+1

2

(9)

Is y+2x in the saturated ideal of sfrc?

IsInSaturatey+2x,sfrc,R

true

(10)

See Also

ChainTools

EqualSaturatedIdeals

Equations

IsIncluded

IsInSaturate

NumberOfSolutions

PolynomialRing

RegularChains

Triangularize

 


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