LimitPoints - Maple Help

RegularChains[AlgebraicGeometryTools]

 LimitPoints
 compute the limit points of a regular chain

 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

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$$\mathrm{with}\left(\mathrm{ChainTools}\right):$$\mathrm{with}\left(\mathrm{AlgebraicGeometryTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[x,y,z\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)
 > $\mathrm{rc}≔\mathrm{Chain}\left(\left[{y}^{5}-{z}^{4},xz-{y}^{2}\right],\mathrm{Empty}\left(R\right),R\right)$
 ${\mathrm{rc}}{≔}{\mathrm{regular_chain}}$ (2)
 > $\mathrm{Display}\left(\mathrm{rc},R\right)$
 $\left\{\begin{array}{cc}{z}{}{x}{-}{{y}}^{{2}}{=}{0}& {}\\ {{y}}^{{5}}{-}{{z}}^{{4}}{=}{0}& {}\\ {z}{\ne }{0}& {}\end{array}\right\$ (3)
 > $\mathrm{lm}≔\mathrm{LimitPoints}\left(\mathrm{rc},R\right)$
 ${\mathrm{lm}}{≔}\left[{\mathrm{regular_chain}}\right]$ (4)
 > $\mathrm{Display}\left(\mathrm{lm},R\right)$
 $\left[\left\{\begin{array}{cc}{x}{=}{0}& {}\\ {y}{=}{0}& {}\\ {z}{=}{0}& {}\end{array}\right\\right]$ (5)
 > $\mathrm{rc}≔\mathrm{Chain}\left(\left[{y}^{5}-{z}^{4}{\left(z+1\right)}^{5},xz{\left(z+1\right)}^{2}-{y}^{2}\right],\mathrm{Empty}\left(R\right),R\right)$
 ${\mathrm{rc}}{≔}{\mathrm{regular_chain}}$ (6)
 > $\mathrm{lm}≔\mathrm{LimitPoints}\left(\mathrm{rc},R\right);$$\mathrm{Display}\left(\mathrm{lm},R\right)$
 ${\mathrm{lm}}{≔}\left[{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right]$
 $\left[\left\{\begin{array}{cc}{x}{=}{0}& {}\\ {y}{=}{0}& {}\\ {z}{=}{0}& {}\end{array}\right\{,}\left\{\begin{array}{cc}{x}{+}{1}{=}{0}& {}\\ {y}{=}{0}& {}\\ {z}{+}{1}{=}{0}& {}\end{array}\right\{,}\left\{\begin{array}{cc}{{x}}^{{4}}{-}{{x}}^{{3}}{+}{{x}}^{{2}}{-}{x}{+}{1}{=}{0}& {}\\ {y}{=}{0}& {}\\ {z}{+}{1}{=}{0}& {}\end{array}\right\\right]$ (7)
 > $\mathrm{lm}≔\mathrm{LimitPoints}\left(\mathrm{rc},R,\left[z\right]\right):$$\mathrm{Display}\left(\mathrm{lm},R\right)$
 $\left[\left\{\begin{array}{cc}{x}{=}{0}& {}\\ {y}{=}{0}& {}\\ {z}{=}{0}& {}\end{array}\right\\right]$ (8)
 > $\mathrm{rc}≔\mathrm{Chain}\left(\left[{y}^{3}-2{y}^{3}+{y}^{2}+{z}^{5},{z}^{4}x+{y}^{3}-{y}^{2}\right],\mathrm{Empty}\left(R\right),R\right)$
 ${\mathrm{rc}}{≔}{\mathrm{regular_chain}}$ (9)
 > $\mathrm{lm}≔\mathrm{LimitPoints}\left(\mathrm{rc},R,\mathrm{coefficient}=\mathrm{complex}\right)$
 ${\mathrm{lm}}{≔}\left[{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right]$ (10)
 > $\mathrm{Display}\left(\mathrm{lm},R\right)$
 $\left[\left\{\begin{array}{cc}{x}{=}{0}& {}\\ {y}{=}{0}& {}\\ {z}{=}{0}& {}\end{array}\right\{,}\left\{\begin{array}{cc}{x}{=}{0}& {}\\ {y}{-}{1}{=}{0}& {}\\ {z}{=}{0}& {}\end{array}\right\\right]$ (11)
 > $\mathrm{lm}≔\mathrm{LimitPoints}\left(\mathrm{rc},R,\mathrm{coefficient}=\mathrm{real}\right)$
 ${\mathrm{lm}}{≔}\left[{\mathrm{regular_semi_algebraic_system}}\right]$ (12)
 > $\mathrm{Display}\left(\mathrm{lm},R\right)$
 $\left[\left\{\begin{array}{cc}{x}{=}{0}& {}\\ {y}{-}{1}{=}{0}& {}\\ {z}{=}{0}& {}\end{array}\right\\right]$ (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.