RegularChains[AlgebraicGeometryTools] - Maple Programming Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Factorization and Solving Equations : RegularChains : AlgebraicGeometryTools Subpackage : RegularChains/AlgebraicGeometryTools/IntersectionMultiplicity

RegularChains[AlgebraicGeometryTools]

  

IntersectionMultiplicity

  

compute the intersection multiplicity of a space curve at a point

 

Calling Sequence

Parameters

Description

Examples

References

Compatibility

Calling Sequence

IntersectionMultiplicity(rc,F,R)

Parameters

R

-

polynomial ring

rc

-

regular chain of R

F

-

list of polynomials of R

Description

• 

The command IntersectionMultiplicity('rc','F','R') returns the intersection multiplicity of the zero-dimensional algebraic variety defined by F at every point of that variety which is a root of the regular chain rc.

• 

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 rc, 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.

• 

It is assumed that rc is a zero-dimensional regular chain, the zero set of which is contained in that of F.

• 

Unless n is equal to 2, the underlying algorithm may fail to compute the multiplicity of certain points of the zero set of rc. 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)

decTriangularizeF,R

decregular_chain,regular_chain,regular_chain,regular_chain

(4)

Displaydec,R

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

(5)

mapIntersectionMultiplicity,dec,F,R

1,regular_chain,2,regular_chain,2,regular_chain,2,regular_chain

(6)

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][IntersectionMultiplicity] command was introduced in Maple 2020.

• 

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

See Also

Display

PolynomialRing

RegularChains

Triangularize