 TriangularizeWithMultiplicity - Maple Help

RegularChains[AlgebraicGeometryTools]

 TriangularizeWithMultiplicity
 compute a triangular decomposition with multiplicities Calling Sequence TriangularizeWithMultiplicity(rc,F,R) Parameters

 R - polynomial ring rc - regular chain of R F - list of polynomials of R Description

 • 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](..). 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)
 > $R≔\mathrm{PolynomialRing}\left(\left[x,y,z\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (2)
 > $F≔\left[{x}^{2}+y+z-1,{y}^{2}+x+z-1,{z}^{2}+x+y-1\right]$
 ${F}{≔}\left[{{x}}^{{2}}{+}{y}{+}{z}{-}{1}{,}{{y}}^{{2}}{+}{x}{+}{z}{-}{1}{,}{{z}}^{{2}}{+}{x}{+}{y}{-}{1}\right]$ (3)
 > $\mathrm{dec}≔\mathrm{TriangularizeWithMultiplicity}\left(F,R\right)$
 ${\mathrm{dec}}{≔}\left[\left[{1}{,}{\mathrm{regular_chain}}\right]{,}\left[{2}{,}{\mathrm{regular_chain}}\right]{,}\left[{2}{,}{\mathrm{regular_chain}}\right]{,}\left[{2}{,}{\mathrm{regular_chain}}\right]\right]$ (4)
 > $\mathrm{Display}\left(\mathrm{dec},R\right)$
 $\left[\left[{1}{,}\left\{\begin{array}{cc}{x}{-}{z}{=}{0}& {}\\ {y}{-}{z}{=}{0}& {}\\ {{z}}^{{2}}{+}{2}{}{z}{-}{1}{=}{0}& {}\end{array}\right\\right]{,}\left[{2}{,}\left\{\begin{array}{cc}{x}{=}{0}& {}\\ {y}{=}{0}& {}\\ {z}{-}{1}{=}{0}& {}\end{array}\right\\right]{,}\left[{2}{,}\left\{\begin{array}{cc}{x}{=}{0}& {}\\ {y}{-}{1}{=}{0}& {}\\ {z}{=}{0}& {}\end{array}\right\\right]{,}\left[{2}{,}\left\{\begin{array}{cc}{x}{-}{1}{=}{0}& {}\\ {y}{=}{0}& {}\\ {z}{=}{0}& {}\end{array}\right\\right]\right]$ (5) References

 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. Compatibility

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