TangentCone - Maple Help

RegularChains[AlgebraicGeometryTools]

 TangentCone
 compute the tangent cone of a space curve at a point given by a regular chain

 Calling Sequence TangentCone(rc, F, R) TangentCone(rc, F, R, vars, outopt)

Parameters

 R - polynomial ring rc - regular chain of R F - list of polynomials of R vars - (optional) list of variable names outopt - (optional) equation of the form output = out, where out is either equations or slopes

Description

 • The command TangentCone(rc, F, R) returns the tangent cone of the space curve defined by F at every point of F defined by rc.
 • By default, the result is a list of pairs [E, ts] where ts is a zero-dimensional regular chain the zero set of which is contained in that of rc, and E is a list of polynomials the zero set of which defines the tangent cone of F at ts.
 • It is assumed that rc is a zero-dimensional regular chain.
 • It is assumed that F generates a one-dimensional ideal and F consists of n-1 polynomials where n is the number of variables in R.
 • The output option, if present, specifies whether the equations, or just the slopes, of the tangent cone are desired. By default, the equations defining the tangent cone are returned, as described above. This can be requested explicitly by including the option output = equations. If the option output = slopes is used, then the result is still a list of pairs [E, ts] with the same interpretation of ts, but now E is a list of polynomials the zero set of which defines the directions of the tangent cone.
 • If the argument vars is present then those unassigned variables are used for the coordinate system of the tangent cone. Otherwise, those coordinates are obtained by prefixing the variables of R with a character when generating the equations. This character is an underscore if the equations of the tangent cone are requested, or a percent character if the slopes are requested. On output, symbols starting with a percent character are typeset in a grayed out color.
 • It is assumed that vars is a list of n unassigned variables.
 • The implementation is based on the method proposed in the paper "A Standard Basis Free Algorithm for Computing the Tangent Cones of a Space Curve" by Parisa Alvandi, Marc Moreno Maza, Eric Schost, Paul Vrbik.
 • This command is part of the RegularChains[AlgebraicGeometryTools] package, so it can be used in the form TangentCone(..) 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][TangentCone](..).

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)
 > $\mathrm{rc}≔\mathrm{Empty}\left(R\right)$
 ${\mathrm{rc}}{≔}{\mathrm{regular_chain}}$ (2)
 > $\mathrm{rc}≔\mathrm{Chain}\left(\left[z-1,y,x\right],\mathrm{rc},R\right)$
 ${\mathrm{rc}}{≔}{\mathrm{regular_chain}}$ (3)
 > $\mathrm{Equations}\left(\mathrm{rc},R\right)$
 $\left[{x}{,}{y}{,}{z}{-}{1}\right]$ (4)
 > $F≔\left[{x}^{2}+{y}^{2}+{z}^{2}-1,{x}^{2}-{y}^{2}-z\left(z-1\right)\right]:$

Determine the equations of the tangent cone to F at the point determined by rc. Express the result in terms of the variables _x, _y, and _z.

 > $\mathrm{tc}≔\mathrm{TangentCone}\left(\mathrm{rc},F,R\right)$
 ${\mathrm{tc}}{≔}\left\{\left[\left[{\mathrm{_z}}{-}{1}{,}{3}{}{{\mathrm{_x}}}^{{2}}{-}{{\mathrm{_y}}}^{{2}}\right]{,}{\mathrm{regular_chain}}\right]\right\}$ (5)

This command does the same in terms of X, Y, and Z. The option output = equations just selects the default behavior explicitly (it has no effect here).

 > $\mathrm{tc}≔\mathrm{TangentCone}\left(\mathrm{rc},F,R,\left[X,Y,Z\right],\mathrm{output}=\mathrm{equations}\right)$
 ${\mathrm{tc}}{≔}\left\{\left[\left[{Z}{-}{1}{,}{3}{}{{X}}^{{2}}{-}{{Y}}^{{2}}\right]{,}{\mathrm{regular_chain}}\right]\right\}$ (6)

Determine the slopes of the same tangent cone. By default, this is returned in terms of the variables %x, %y, and %z. (These are typeset as grayed out versions of x, y, and z.)

 > $\mathrm{tc}≔\mathrm{TangentCone}\left(\mathrm{rc},F,R,\mathrm{output}=\mathrm{slopes}\right)$
 ${\mathrm{tc}}{≔}\left\{\left[\left[{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%z}}\right]\right){,}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%y}}\right]\right){-}{1}{,}{3}{}{{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%x}}\right]\right)}^{{2}}{-}{1}\right]{,}{\mathrm{regular_chain}}\right]{,}\left[\left[{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%z}}\right]\right){,}{{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%y}}\right]\right)}^{{2}}{-}{3}{,}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%x}}\right]\right){-}{1}\right]{,}{\mathrm{regular_chain}}\right]\right\}$ (7)
 > $\mathrm{tc}≔\mathrm{TangentCone}\left(\mathrm{rc},F,R,\left[X,Y,Z\right],\mathrm{output}=\mathrm{slopes}\right)$
 ${\mathrm{tc}}{≔}\left\{\left[\left[{Z}{,}{Y}{-}{1}{,}{3}{}{{X}}^{{2}}{-}{1}\right]{,}{\mathrm{regular_chain}}\right]{,}\left[\left[{Z}{,}{{Y}}^{{2}}{-}{3}{,}{X}{-}{1}\right]{,}{\mathrm{regular_chain}}\right]\right\}$ (8)

References

 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][TangentCone] command was introduced in Maple 2020.