Cylindrify - Maple Help
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RegularChains[AlgebraicGeometryTools]

  

Cylindrify

  

Simplify a polynomial system in the local ring of a point

 

Calling Sequence

Parameters

Description

Examples

References

Compatibility

Calling Sequence

Cylindrify(rc,F, R)

Parameters

R

-

polynomial ring

rc

-

regular chain of R

F

-

list of polynomials of R

Description

• 

The command Cylindrify(rc,F, R) returns a list of polynomials G such that F and G have the same intersection multiplicity at every point defined by the zero-dimensional regular chain rc. Moreover, either G is F itself or there exists a variable v of R and a polynomial g of G such that:

1. 

the polynomial g has degree one in v and its leading coefficient in v is invertible in the local ring at p for every point p defined by the zero-dimensional regular chain rc; and

2. 

each other polynomial in G is independent of v.

• 

In that latter case, the polynomial set G facilitates the study of the local properties of the zero set of F around every point solving rc.

• 

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.

• 

This is not a complete algorithm: in some rare cases, the command will signal an error and fail.

• 

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

Examples

(1)

(2)

(3)

(4)

(5)

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

• 

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

See Also

Display

IsTransverse

PolynomialRing

RegularChains

Triangularize

 


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