compute the limit points of a regular chain
LimitPoints(rc, R, L)
LimitPoints(rc, R, coefficient=real)
LimitPoints(rc, R, output=rootof)
LimitPoints(rc, R, output=chain)
regular chain of R
list of polynomials of R
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](..).
R ≔ PolynomialRing⁡x,y,z
rc ≔ Chain⁡y5−z4,x⁢z−y2,Empty⁡R,R
lm ≔ LimitPoints⁡rc,R
rc ≔ Chain⁡y5−z4⁢z+15,x⁢z⁢z+12−y2,Empty⁡R,R
lm ≔ LimitPoints⁡rc,R;Display⁡lm,R
lm ≔ LimitPoints⁡rc,R,z:Display⁡lm,R
rc ≔ Chain⁡y3−2⁢y3+y2+z5,z4⁢x+y3−y2,Empty⁡R,R
lm ≔ LimitPoints⁡rc,R,coefficient=complex
lm ≔ LimitPoints⁡rc,R,coefficient=real
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)
The RegularChains[AlgebraicGeometryTools][LimitPoints] command was introduced in Maple 2020.
For more information on Maple 2020 changes, see Updates in Maple 2020.
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