The optional arguments are passed to the IntersectionMultiplicity command. For a full description, see IntersectionMultiplicity.

The command TriangularizeWithMultiplicity(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 $\left[m\,t\right]$ where $t$ 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 system of equations defined by $F$ at every point defined by $t$.

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

Unless $n=2$, the underlying algorithm may fail to compute the multiplicity of certain points of the zero set of $F$. When this occurs, $m$ is usually set to $\mathrm{FAIL}$; see IntersectionMultiplicity for more details.

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)$

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

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

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

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

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

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

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

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

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

[2] 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 2015), Lecture Notes in Computer Science 9301, (2015), 45-60.

[3] M. Moreno Maza and R. Sandford. Towards Extending Fulton's Intersection Multiplicity Algorithm Beyond the Bivariate Case. Computer Algebra in Scientific Computing (CASC 2021), Lecture Notes in Computer Science 12865, (2021), 232-251.

The maxdepth, maxshift and method options were added in Maple 2022. method=tangentcone corresponds to the algorithm in Maple 2020 and 2021.

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

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