TriangularizeWithMultiplicity - Maple Help

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RegularChains[AlgebraicGeometryTools]

  

TriangularizeWithMultiplicity

  

compute a triangular decomposition with multiplicities

 

Calling Sequence

Parameters

Description

Examples

References

Compatibility

Calling Sequence

TriangularizeWithMultiplicity(rc,F,R)

Parameters

R

-

polynomial ring

rc

-

regular chain of R

F

-

list of polynomials of R

Description

• 

The command TriangularizeWithMultiplicity('rc','F','R') returns a triangular decomposition of the zero set of F together with the multiplicity of every point of that zero set.

• 

The result is a list of pairs [m,ts] where ts is a zero-dimensional regular chain the zero set of which is contained in that of F, and m is the intersection multiplicity of the space curve defined by F at every point defined by ts.

• 

It is assumed that F generates a zero-dimensional ideal and F consists of n polynomials where n is the number of variables in R.

• 

Unless n is equal to 2, the underlying algorithm may fail to compute the multiplicity of certain points of the zero set of F. In this case, an error is signaled.

• 

The implementation is based on the method proposed in the paper "On Fulton's Algorithm for Computing Intersection Multiplicities" by Steffen Marcus, Marc Moreno Maza, Paul Vrbik.

• 

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

Examples

withRegularChains:withAlgebraicGeometryTools

Cylindrify,IntersectionMultiplicity,IsTransverse,LimitPoints,RationalFunctionLimit,RegularChainBranches,TangentCone,TangentPlane,TriangularizeWithMultiplicity

(1)

RPolynomialRingx,y,z

Rpolynomial_ring

(2)

Fx2+y+z1,y2+x+z1,z2+x+y1

Fx2+y+z1,y2+x+z1,z2+x+y1

(3)

decTriangularizeWithMultiplicityF,R

dec1,regular_chain,2,regular_chain,2,regular_chain,2,regular_chain

(4)

Displaydec,R

1,xz=0yz=0z2+2z1=0,2,x=0y=0z1=0,2,x=0y1=0z=0,2,x1=0y=0z=0

(5)

References

  

Steffen Marcus, Marc Moreno Maza, Paul Vrbik "On Fulton's Algorithm for Computing Intersection Multiplicities." Computer Algebra in Scientific Computing (CASC), Lecture Notes in Computer Science - 7442, (2012): 198-211.

  

Parisa Alvandi, Marc Moreno Maza, Eric Schost, Paul Vrbik "A Standard Basis Free Algorithm for Computing the Tangent Cones of a Space Curve." Computer Algebra in Scientific Computing (CASC), Lecture Notes in Computer Science - 9301, (2015): 45-60.

Compatibility

• 

The RegularChains[AlgebraicGeometryTools][TriangularizeWithMultiplicity] command was introduced in Maple 2020.

• 

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

See Also

Display

IntersectionMultiplicity

PolynomialRing

RegularChains

Triangularize