compute a triangular decomposition with multiplicities
list of polynomials with integer coefficients
polynomial ring of characteristic zero
sequence of optional equations of the form keyword=value, where keyword is maxdepth, maxshift, or method
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 m,t 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 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](..).
R ≔ PolynomialRing⁡x,y,z:
F ≔ x2+y+z−1,y2+x+z−1,z2+x+y−1
dec ≔ TriangularizeWithMultiplicity⁡F,R
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 −1+2,−1+2,−1+2 and −1−2,−1−2,−1−2.
 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.
 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.
 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.
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