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Calling Sequence
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TriangularizeWithMultiplicity(F,R)
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Parameters
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F
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list of polynomials with integer coefficients
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R
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polynomial ring of characteristic zero
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options
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sequence of optional equations of the form keyword=value, where keyword is maxdepth, maxshift, or method
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Options
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The optional arguments are passed to the IntersectionMultiplicity command. For a full description, see IntersectionMultiplicity.
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Description
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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.
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The result is a list of pairs where is a zero-dimensional regular chain the zero set of which is contained in that of , and is the intersection multiplicity of the system of equations defined by at every point defined by .
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It is assumed that consists of polynomials generating a zero-dimensional ideal, where is the number of variables in .
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Unless , the underlying algorithm may fail to compute the multiplicity of certain points of the zero set of . When this occurs, is usually set to ; see IntersectionMultiplicity for more details.
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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](..).
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Examples
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>
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>
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>
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>
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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 and .
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References
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[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.
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[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.
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[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.
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Compatibility
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The maxdepth, maxshift and method options were added in Maple 2022. method=tangentcone corresponds to the algorithm in Maple 2020 and 2021.
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The RegularChains[AlgebraicGeometryTools][TriangularizeWithMultiplicity] command was introduced in Maple 2020.
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