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](..).

$\mathrm{with}\left(\mathrm{RegularChains}\right)\:$$\mathrm{with}\left(\mathrm{ChainTools}\right)\:$$\mathrm{with}\left(\mathrm{AlgebraicGeometryTools}\right)\:$

$R≔\mathrm{PolynomialRing}\left(\left[x\,y\,z\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$\mathrm{rc}≔\mathrm{Chain}\left(\left[y^5-z^4\,xz-y^2\right]\,\mathrm{Empty}\left(R\right)\,R\right)$

$\mathrm{rc}≔\mathrm{regular\_chain}$

$\mathrm{Display}\left(\mathrm{rc}\,R\right)$

$\left\{\begin{array}{cc}zx-y^2=0& \\ y^5-z^4=0& \\ z\ne 0& \end{array}\right.$

$\mathrm{lm}≔\mathrm{LimitPoints}\left(\mathrm{rc}\,R\right)$

$\mathrm{lm}≔\left[\mathrm{regular\_chain}\right]$

$\mathrm{Display}\left(\mathrm{lm}\,R\right)$

$\left[\left\{\begin{array}{cc}x=0& \\ y=0& \\ z=0& \end{array}\right.\right]$

$\mathrm{rc}≔\mathrm{Chain}\left(\left[y^5-z^4{\left(z+1\right)}^5\,xz{\left(z+1\right)}^2-y^2\right]\,\mathrm{Empty}\left(R\right)\,R\right)$

$\mathrm{rc}≔\mathrm{regular\_chain}$

$\mathrm{lm}≔\mathrm{LimitPoints}\left(\mathrm{rc}\,R\right)\;$$\mathrm{Display}\left(\mathrm{lm}\,R\right)$

$\mathrm{lm}≔\left[\mathrm{regular\_chain}\,\mathrm{regular\_chain}\,\mathrm{regular\_chain}\right]$

$\left[\left\{\begin{array}{cc}x=0& \\ y=0& \\ z=0& \end{array}\right.\,\left\{\begin{array}{cc}x+1=0& \\ y=0& \\ z+1=0& \end{array}\right.\,\left\{\begin{array}{cc}{x}^4-x^3+x^2-x+1=0& \\ y=0& \\ z+1=0& \end{array}\right.\right]$

$\mathrm{lm}≔\mathrm{LimitPoints}\left(\mathrm{rc}\,R\,\left[z\right]\right)\:$$\mathrm{Display}\left(\mathrm{lm}\,R\right)$

$\left[\left\{\begin{array}{cc}x=0& \\ y=0& \\ z=0& \end{array}\right.\right]$

$\mathrm{rc}≔\mathrm{Chain}\left(\left[y^3-2y^3+y^2+z^5\,z^{4}x+y^3-y^2\right]\,\mathrm{Empty}\left(R\right)\,R\right)$

$\mathrm{rc}≔\mathrm{regular\_chain}$

$\mathrm{lm}≔\mathrm{LimitPoints}\left(\mathrm{rc}\,R\,\mathrm{coefficient}=\mathrm{complex}\right)$

$\mathrm{lm}≔\left[\mathrm{regular\_chain}\,\mathrm{regular\_chain}\right]$

$\mathrm{Display}\left(\mathrm{lm}\,R\right)$

$\left[\left\{\begin{array}{cc}x=0& \\ y=0& \\ z=0& \end{array}\right.\,\left\{\begin{array}{cc}x=0& \\ y-1=0& \\ z=0& \end{array}\right.\right]$

$\mathrm{lm}≔\mathrm{LimitPoints}\left(\mathrm{rc}\,R\,\mathrm{coefficient}=\mathrm{real}\right)$

$\mathrm{lm}≔\left[\mathrm{regular\_semi\_algebraic\_system}\right]$

$\mathrm{Display}\left(\mathrm{lm}\,R\right)$

$\left[\left\{\begin{array}{cc}x=0& \\ y-1=0& \\ z=0& \end{array}\right.\right]$

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.