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RegularChains[AlgebraicGeometryTools]

  

LimitPoints

  

compute the limit points of a regular chain

 

Calling Sequence

Parameters

Description

Examples

References

Compatibility

Calling Sequence

LimitPoints(rc, R)

LimitPoints(rc, R, L)

LimitPoints(rc, R, coefficient=real)

LimitPoints(rc, R, output=rootof)

LimitPoints(rc, R, output=chain)

Parameters

R

-

polynomial ring

rc

-

regular chain of R

L

-

list of polynomials of R

Description

• 

The command LimitPoints(rc, R) returns the non-trivial limit points of the  quasi-component given by the regular chain rc in the Zariski topology. Non-trivial refers to the limit points that are not points of that same quasi-component.

• 

The returned limit points forum a zero-dimensional variety which, by default, is given as the union of the zero sets of regular chains.

• 

It is assumed that the coefficient field of R is the field of rational numbers.

• 

It is assumed that rc is a one-dimensional strongly normalized regular chain.

• 

This implies that every initial of a polynomial f in rc is either constant or univariate in a variable, say v, of R which is not algebraic w.r.t. rc.

• 

It is assumed that the polynomials in L are univariate in that variable v. Moreover, it is assumed that every root of a polynomial in L is a root of the initial of a polynomial in rc.

• 

Each limit point returned by LimitPoints(rc, R) is obtained by following a branch (given by a Puiseux series solution) of rc associated with a root of the product of the initials of rc.

• 

If the optional argument L is present, then only the limit points obtained from a branch associated with a root of a polynomial in L are returned.

• 

If the option coefficient=real is present, then only the points obtained from a real branch are returned.

• 

If the option output=chain is present, then the returned limit points are given as solutions of zero-dimensional regular chains; this is the default representation for the returned limit points.

• 

If the option output=rootof is present, then RootOf expressions are are used (instead of regular chains) to represent the coordinates of the limit points.

• 

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

Examples

withRegularChains:withChainTools:withAlgebraicGeometryTools:

RPolynomialRingx,y,z

Rpolynomial_ring

(1)

rcChainy5z4,xzy2,EmptyR,R

rcregular_chain

(2)

Displayrc,R

zxy2=0y5z4=0z0

(3)

lmLimitPointsrc,R

lmregular_chain

(4)

Displaylm,R

x=0y=0z=0

(5)

rcChainy5z4z+15,xzz+12y2,EmptyR,R

rcregular_chain

(6)

lmLimitPointsrc,R;Displaylm,R

lmregular_chain,regular_chain,regular_chain

x=0y=0z=0,x+1=0y=0z+1=0,x4x3+x2x+1=0y=0z+1=0

(7)

lmLimitPointsrc,R,z:Displaylm,R

x=0y=0z=0

(8)

rcChainy32y3+y2+z5,z4x+y3y2,EmptyR,R

rcregular_chain

(9)

lmLimitPointsrc,R,coefficient=complex

lmregular_chain,regular_chain

(10)

Displaylm,R

x=0y=0z=0,x=0y1=0z=0

(11)

lmLimitPointsrc,R,coefficient=real

lmregular_semi_algebraic_system

(12)

Displaylm,R

x=0y1=0z=0

(13)

References

  

Parisa Alvandi, Changbo Chen, Marc Moreno Maza "Computing the Limit Points of the Quasi-component of a Regular Chain in Dimension One." Computer Algebra in Scientific Computing (CASC), Lecture Notes in Computer Science - 8136, (2013): 30-45.

  

Parisa Alvandi, Masoud Ataei, Mahsa Kazemi, Marc Moreno Maza "On the Extended Hensel Construction and its application to the computation of real limit points." J. Symb. Comput. 98: 120-162 (2020)

Compatibility

• 

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

• 

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

See Also

Chain

Display

Empty

Equations

PolynomialRing

RegularChainBranches

RegularChains