The command TriangularizeWithMultiplicity('rc','F','R') returns a triangular decomposition of the zero set of F together with the multiplicity of every point of that zero set.

The result is a list of pairs [m,ts] where ts is a zero-dimensional regular chain the zero set of which is contained in that of F, and m is the intersection multiplicity of the space curve defined by F at every point defined by ts.

It is assumed that F generates a zero-dimensional ideal and F consists of n polynomials where n is the number of variables in R.

Unless n is equal to 2, the underlying algorithm may fail to compute the multiplicity of certain points of the zero set of F. In this case, an error is signaled.

The implementation is based on the method proposed in the paper "On Fulton's Algorithm for Computing Intersection Multiplicities" by Steffen Marcus, Marc Moreno Maza, Paul Vrbik.

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

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

$\left[\mathrm{Cylindrify}\,\mathrm{IntersectionMultiplicity}\,\mathrm{IsTransverse}\,\mathrm{LimitPoints}\,\mathrm{RationalFunctionLimit}\,\mathrm{RegularChainBranches}\,\mathrm{TangentCone}\,\mathrm{TangentPlane}\,\mathrm{TriangularizeWithMultiplicity}\right]$

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

$R≔\mathrm{polynomial\_ring}$

$F≔\left[x^2+y+z-1\,y^2+x+z-1\,z^2+x+y-1\right]$

$F≔\left[x^2+y+z-1\,y^2+x+z-1\,z^2+x+y-1\right]$

$\mathrm{dec}≔\mathrm{TriangularizeWithMultiplicity}\left(F\,R\right)$

$\mathrm{dec}≔\left[\left[1\,\mathrm{regular\_chain}\right]\,\left[2\,\mathrm{regular\_chain}\right]\,\left[2\,\mathrm{regular\_chain}\right]\,\left[2\,\mathrm{regular\_chain}\right]\right]$

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

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

Steffen Marcus, Marc Moreno Maza, Paul Vrbik "On Fulton's Algorithm for Computing Intersection Multiplicities." Computer Algebra in Scientific Computing (CASC), Lecture Notes in Computer Science - 7442, (2012): 198-211.

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.

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

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