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/IsTransverse

RegularChains[AlgebraicGeometryTools]

  

IsTransverse

  

check whether an hypersurface and a space curve meet transversely

 

Calling Sequence

Parameters

Description

Examples

References

Compatibility

Calling Sequence

IsTransverse(rc,f, F, R)

Parameters

R

-

polynomial ring

rc

-

regular chain of R

f

-

a polynomial of R

F

-

list of polynomials of R

Description

• 

The command IsTransverse(rc,f, F, R) returns true if and only if the hypersurface defined by f and the space curve defined by F meet transversely at every point defined by the zero-dimensional regular chain rc.

• 

In other words, this command returns true if and only if the hypersurface defined by f and the tangent cone of the space curve defined by F at p intersect at p, and only at p, in a neighbourhood of p, for every point p defined by the regular chain rc.

• 

It is assumed that rc is a zero-dimensional regular chain.

• 

It is assumed that F generates a one-dimensional ideal and F consists of n-1 polynomials where n is the number of variables in R.

• 

It is assumed that the hypersurface defined by f is non-singular at every point defined by rc.

• 

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

Examples

withRegularChains:withChainTools:withAlgebraicGeometryTools:

RPolynomialRingx,y,z

Rpolynomial_ring

(1)

Fx,xy2z2,yz3

Fx,y2z2+x,z3+y

(2)

decTriangularizeF,R

decregular_chain,regular_chain

(3)

Displaydec,R

x=0yz3=0z4+1=0,x=0y=0z=0

(4)

IsTransversedec1,F3,F1..2,R

true

(5)

IsTransversedec2,F3,F1..2,R

true

(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][IsTransverse] command was introduced in Maple 2020.

• 

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

See Also

Display

PolynomialRing

RegularChains

Triangularize