TriangularizeWithMultiplicity - Maple Help
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RegularChains[AlgebraicGeometryTools]

 TriangularizeWithMultiplicity
 compute a triangular decomposition with multiplicities

 Calling Sequence TriangularizeWithMultiplicity(F,R)

Parameters

 F - list of polynomials with integer coefficients R - polynomial ring of characteristic zero options - sequence of optional equations of the form keyword=value, where keyword is maxdepth, maxshift, or method

Options

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

Description

 • 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](..).

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$$\mathrm{with}\left(\mathrm{AlgebraicGeometryTools}\right)$
 $\left[{\mathrm{Cylindrify}}{,}{\mathrm{IntersectionMultiplicity}}{,}{\mathrm{IsTransverse}}{,}{\mathrm{LimitPoints}}{,}{\mathrm{RationalFunctionLimit}}{,}{\mathrm{RegularChainBranches}}{,}{\mathrm{TangentCone}}{,}{\mathrm{TangentPlane}}{,}{\mathrm{TriangularizeWithMultiplicity}}\right]$ (1)
 > $R≔\mathrm{PolynomialRing}\left(\left[x,y,z\right]\right):$
 > $F≔\left[{x}^{2}+y+z-1,{y}^{2}+x+z-1,{z}^{2}+x+y-1\right]$
 ${F}{≔}\left[{{x}}^{{2}}{+}{y}{+}{z}{-}{1}{,}{{y}}^{{2}}{+}{x}{+}{z}{-}{1}{,}{{z}}^{{2}}{+}{x}{+}{y}{-}{1}\right]$ (2)
 > $\mathrm{dec}≔\mathrm{TriangularizeWithMultiplicity}\left(F,R\right)$
 ${\mathrm{dec}}{≔}\left[\left[{1}{,}{\mathrm{regular_chain}}\right]{,}\left[{2}{,}{\mathrm{regular_chain}}\right]{,}\left[{2}{,}{\mathrm{regular_chain}}\right]{,}\left[{2}{,}{\mathrm{regular_chain}}\right]\right]$ (3)
 > $\mathrm{Display}\left(\mathrm{dec},R\right)$
 $\left[\left[{1}{,}\left\{\begin{array}{cc}{x}{-}{z}{=}{0}& {}\\ {y}{-}{z}{=}{0}& {}\\ {{z}}^{{2}}{+}{2}{}{z}{-}{1}{=}{0}& {}\end{array}\right\\right]{,}\left[{2}{,}\left\{\begin{array}{cc}{x}{=}{0}& {}\\ {y}{=}{0}& {}\\ {z}{-}{1}{=}{0}& {}\end{array}\right\\right]{,}\left[{2}{,}\left\{\begin{array}{cc}{x}{=}{0}& {}\\ {y}{-}{1}{=}{0}& {}\\ {z}{=}{0}& {}\end{array}\right\\right]{,}\left[{2}{,}\left\{\begin{array}{cc}{x}{-}{1}{=}{0}& {}\\ {y}{=}{0}& {}\\ {z}{=}{0}& {}\end{array}\right\\right]\right]$ (4)

Here, calling TriangularizeWithMultiplicity returns four regular chains and the intersection multiplicities corresponding to each point encoded in the regular chain. Moreover, while the last 3 regular chains encode just a point, the first regular chain encodes two points, namely $\left(-1+\sqrt{2},-1+\sqrt{2},-1+\sqrt{2}\right)$ and $\left(-1-\sqrt{2},-1-\sqrt{2},-1-\sqrt{2}\right)$.

References

 [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.

Compatibility

 • 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.