RegularChains[ChainTools] - Maple Programming Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Factorization and Solving Equations : RegularChains : ChainTools Subpackage : RegularChains/ChainTools/ChangeOfCoordinates

RegularChains[ChainTools]

  

ChangeOfCoordinates

  

change of coordinate system for a regular chain

 

Calling Sequence

Parameters

Description

Examples

References

Compatibility

Calling Sequence

ChangeOfCoordinates(rc, R, M, V)

Parameters

rc

-

regular chain of R

R

-

polynomial ring

M

-

matrix

V

-

variable list

Description

• 

The command ChangeOfCoordinates returns a list dec2 of regular chains of R2 which forms a Kalkbrener decomposition of the same saturated ideal as rc, after applying to this ideal the linear change of coordinates defined by M and V.

• 

This linear change of coordinates maps the coordinates (given by the variables of the polynomial ring R) to the ordered variables given by V via the invertible linear transformation given by M.

• 

In cases where the change of coordinates is a permutation, it can also be performed by the ChangeOfOrder command.

• 

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

Examples

withRegularChains:withChainTools:

F5y43,20xy+z,x5y5+3y+1

F5y43,20xy+z,x5y5+3y+1

(1)

varsz,x,y

varsz,x,y

(2)

In the example below, we consider a change of coordinates which is given by a permutation matrix, hence, this example is a change of variable order.

vars2x,y,z

vars2x,y,z

(3)

MMatrix0,1,0,0,0,1,1,0,0

M010001100

(4)

RPolynomialRingvars

Rpolynomial_ring

(5)

decTriangularizeF,R,normalized=yes

decregular_chain

(6)

rcdec1

rcregular_chain

(7)

Displayrc,R

z+20xy=05x5+12y+5=05y43=0

(8)

dec2ChangeOfCoordinatesrc,R,M,vars2

dec2regular_chain,polynomial_ring

(9)

rc2dec21

rc2regular_chain

(10)

R2dec22

R2polynomial_ring

(11)

Displayrc2,R2

12x+5z5+5=012y+5z5+240z+5=03125z20+12500z15+18750z10+12500z559083=0

(12)

dec0ChangeOfCoordinatesrc2,R2,LinearAlgebra:-MatrixInverseM,vars

dec0regular_chain,polynomial_ring

(13)

rc0dec01

rc0regular_chain

(14)

Displayrc0,R

z+20xy=05x5+12y+5=05y43=0

(15)

EqualSaturatedIdealsrc,rc0,R

true

(16)

Because this change of coordinates is a change of variable order, it can also be performed by the ChangeOfOrder command.

rc3ChangeOfOrderrc,R,R2

rc3regular_chain

(17)

Displayrc3,R2

12x+5z5+5=012y+5z5+240z+5=03125z20+12500z15+18750z10+12500z559083=0

(18)

EqualSaturatedIdealsrc2,rc3,R2

true

(19)

In the next example, we consider a change of coordinates which is not a change of variable order.

MMatrix0,1,0,1,0,1,0,0,1

M01010−1001

(20)

vars2X,Y,Z

vars2X,Y,Z

(21)

RPolynomialRingvars

Rpolynomial_ring

(22)

decTriangularizeF,R,normalized=yes

decregular_chain

(23)

rcdec1

rcregular_chain

(24)

Displayrc,R

z+20xy=05x5+12y+5=05y43=0

(25)

dec2ChangeOfCoordinatesrc,R,M,vars2

dec2regular_chain,polynomial_ring

(26)

rc2dec21

rc2regular_chain

(27)

R2dec22

R2polynomial_ring

(28)

Displayrc2,R2

20X+Y21Z=05Y525ZY4+50Z2Y350Z3Y2+15Y38400003Z16000000=05Z43=0

(29)

dec0ChangeOfCoordinatesrc2,R2,LinearAlgebra:-MatrixInverseM,vars

dec0regular_chain,polynomial_ring

(30)

rc0dec01

rc0regular_chain

(31)

Displayrc0,R

z+20xy=05x5+12y+5=05y43=0

(32)

EqualSaturatedIdealsrc,rc0,R

true

(33)

References

  

Boulier, F.; Lemaire, F. and Moreno Maza, M. "PARDI!." Proc. ISSAC, 2001.

Compatibility

• 

The RegularChains[ChainTools][ChangeOfCoordinates] command was introduced in Maple 2020.

• 

For more information on Maple 2020 changes, see Updates in Maple 2020.

See Also

ChainTools

ChangeOfOrder

EqualSaturatedIdeals

PolynomialRing

Triangularize