compute a pre-comprehensive triangular decomposition - Maple Help

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RegularChains[ParametricSystemTools][PreComprehensiveTriangularize] - compute a pre-comprehensive triangular decomposition

Calling Sequence

PreComprehensiveTriangularize(sys, d, R)

Parameters

sys

-

list of polynomials

d

-

number of parameters

R

-

polynomial ring

Description

• 

The command PreComprehensiveTriangularize(sys, d, R)  returns a pre-comprehensive triangular decomposition of sys, with respect to the last d variables of R.

• 

A pre-comprehensive triangular decomposition is a refined triangular decomposition (in the Lazard sense) with additional properties, aiming at studying parametric polynomial systems.

• 

Let U be the last d variables of R, which we regard as parameters. A finite set S of regular chains of R forms a pre-comprehensive triangular decomposition of F with respect to U, if for every parameter value u, there exists a subset Su of S such that

  

(1) the regular chains of Su specialize well at u, and

  

(2) after specialization at u, these chains form a triangular decomposition (in the Lazard sense) of the polynomial system F specialized at u. See the command DefiningSet for the term specialize well.

Examples

withRegularChains:

withConstructibleSetTools:

withParametricSystemTools:

R:=PolynomialRingx,y,s

R:=polynomial_ring

(1)

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

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

(2)

A pre-comprehensive triangular decomposition of F consists of three regular chains.

pctd:=PreComprehensiveTriangularizeF,1,R

pctd:=regular_chain,regular_chain,regular_chain

(3)

mapInfo,pctd,R

y+1xs,y2+ys,x+1,y+1,s,x,y,s

(4)

Compare it with the output of Triangularize.

dec:=TriangularizeF,R,output=lazard

dec:=regular_chain,regular_chain

(5)

mapInfo,dec,R

y+1xs,y2+ys,x+1,y+1,s

(6)

See Also

ComprehensiveTriangularize, ConstructibleSet, DefiningSet, DiscriminantSet, Info, RegularChains, Triangularize


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