dimension of a regular chain - Maple Help

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RegularChains[ChainTools][Dimension] - dimension of a regular chain

Calling Sequence

Dimension(rc, R)

Parameters

rc

-

regular chain of R

R

-

polynomial ring

Description

• 

The command Dimension(rc, R) returns the dimension of the saturated ideal of rc. This is also the number of variables of R minus the number of elements in rc.

• 

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

Examples

withRegularChains:

withChainTools:

R:=PolynomialRingx,y,a,b,c,d,g,h

R:=polynomial_ring

(1)

sys:=ax+byg,cx+dyh

sys:=ax+byg,cx+dyh

(2)

decl:=Triangularizesys,R,'output'='lazard'

decl:=regular_chain,regular_chain,regular_chain,regular_chain,regular_chain,regular_chain,regular_chain,regular_chain,regular_chain,regular_chain,regular_chain

(3)

mapEquations,decl,R

cx+dyh,dabcyha+cg,cx+dyh,dabc,hbdg,ax+byg,dyh,c,dyh,a,hbdg,c,cxh,hacg,b,d,ax+byg,c,d,h,cx+dy,dabc,g,h,byg,a,c,d,h,y,a,c,g,h,x,b,d,g,h,a,b,c,d,g,h

(4)

We see that RegularChains[Triangularize] produces the regular chains in decreasing order of dimension. This is, in fact, part of the specifications of this function.

mapDimension,decl,R

6,5,5,4,4,4,4,3,3,3,2

(5)

Here is another simple example with a triangular decomposition containing regular chains of different dimensions.

R:=PolynomialRingx,y,z

R:=polynomial_ring

(6)

sys:=xx1x2,xx1y1+x2y,xx1z

sys:=xx1y1+x2y,xx1z,xx1x2

(7)

dec:=Triangularizesys,R

dec:=regular_chain,regular_chain,regular_chain

(8)

mapEquations,dec,R

x,x1,y,x2,y1,z

(9)

mapDimension,dec,R

2,1,0

(10)

These regular chains are a surface, a line, and a point respectively.

See Also

ChainTools, Equations, map, PolynomialRing, RegularChains, Triangularize


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