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RegularChains[ParametricSystemTools]

  

Specialize

  

specialize a list of regular chains at a point

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

Specialize(pt, lrc, R)

Parameters

pt

-

point with coordinates in rational number field or a finite field

lrc

-

list of regular chains

R

-

polynomial ring

Description

• 

The command Specialize(pt, lrc, R) returns a list of regular chains obtained from those of lrc by specialization  at the point pt.

• 

The point pt is given by a list of rational numbers or a list of elements in a finite field; moreover, the number of coordinates in pt must be less than or equal to the number of variables of R.

• 

All polynomials in each regular chain of lrc are evaluated at the  last nopspt variables of R using the corresponding coordinates of pt.

• 

Regular chains in lrc must specialize well at pt, otherwise an error message displays.

• 

This command is part of the RegularChains[ParametricSystemTools] package, so it can be used in the form Specialize(..) only after executing the command with(RegularChains[ParametricSystemTools]). However, it can always be accessed through the long form of the command by using RegularChains[ParametricSystemTools][Specialize](..).

Examples

withRegularChains:

withChainTools:

withConstructibleSetTools:

withParametricSystemTools:

The following example shows how to analyze the output of a comprehensive triangular decomposition.

RPolynomialRingx,y,s

R:=polynomial_ring

(1)

Fsy+1x,sx+1y

F:=sy+1x,sx+1y

(2)

pctd,cellsComprehensiveTriangularizeF,1,R

pctd,cells:=regular_chain,regular_chain,regular_chain,constructible_set,3,2,constructible_set,1

(3)

The first part is a list of regular chains which form a pre-comprehensive triangular decomposition of F. The second part is a partition of the projection image of V(F) to the last coordinate. Each constructible set is associated with indices of regular chains in the first part.

lcsseqcellsi1,i=1..nopscells

lcs:=constructible_set,constructible_set

(4)

Consider a specialization point pts=4.

pt4

pt:=4

(5)

Try to figure out to which partition pt belongs.

liBelongsTopt,lcs,R;ili1

li:=2

i:=2

(6)

Then retrieve the indices of regular chains that specialize well at pt.

indcellsi2

ind:=1

(7)

lrc_indmapi→pctdi,ind

lrc_ind:=regular_chain

(8)

mapInfo,lrc_ind,R

y+1xs,y2+ys

(9)

Thus you know that the regular chains in lrc_ind all specialize well at the point pt. Then you can do simple substitutions.

lrc_spSpecializept,lrc_ind,R

lrc_sp:=regular_chain

(10)

Regular chains of lrc_sp form a triangular decomposition of F after specialization at pt.

mapInfo,lrc_sp,R

y+1x4,y2+y4

(11)

See Also

BelongsTo

ComprehensiveTriangularize

ConstructibleSet

Info

ParametricSystemTools

PreComprehensiveTriangularize

RegularChains

 


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