
Calling Sequence


IntersectionMultiplicity(rc,F,R)


Parameters


R



polynomial ring

rc



regular chain of R

F



list of polynomials of R





Description


•

The command IntersectionMultiplicity('rc','F','R') returns the intersection multiplicity of the zerodimensional algebraic variety defined by F at every point of that variety which is a root of the regular chain rc.

•

The result is a list of pairs [m,ts] where ts is a zerodimensional regular chain the zero set of which is contained in that of rc, 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 zerodimensional ideal and F consists of n polynomials where n is the number of variables in R.

•

It is assumed that rc is a zerodimensional regular chain, the zero set of which is contained in that of F.

•

Unless n is equal to 2, the underlying algorithm may fail to compute the multiplicity of certain points of the zero set of rc. 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) 
>

$R\u2254\mathrm{PolynomialRing}\left(\left[x\,y\,z\right]\right)$

${R}{\u2254}{\mathrm{polynomial\_ring}}$
 (2) 
>

$F\u2254\left[{x}^{2}+y+z1\,{y}^{2}+x+z1\,{z}^{2}+x+y1\right]$

${F}{\u2254}\left[{{x}}^{{2}}{+}{y}{+}{z}{}{1}{\,}{{y}}^{{2}}{+}{x}{+}{z}{}{1}{\,}{{z}}^{{2}}{+}{x}{+}{y}{}{1}\right]$
 (3) 
>

$\mathrm{dec}\u2254\mathrm{Triangularize}\left(F\,R\right)$

${\mathrm{dec}}{\u2254}\left[{\mathrm{regular\_chain}}{\,}{\mathrm{regular\_chain}}{\,}{\mathrm{regular\_chain}}{\,}{\mathrm{regular\_chain}}\right]$
 (4) 
>

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

$\left[\left\{\begin{array}{cc}{x}{}{z}{=}{0}& {}\\ {y}{}{z}{=}{0}& {}\\ {{z}}^{{2}}{+}{2}{}{z}{}{1}{=}{0}& {}\end{array}\right.{\,}\left\{\begin{array}{cc}{x}{=}{0}& {}\\ {y}{=}{0}& {}\\ {z}{}{1}{=}{0}& {}\end{array}\right.{\,}\left\{\begin{array}{cc}{x}{=}{0}& {}\\ {y}{}{1}{=}{0}& {}\\ {z}{=}{0}& {}\end{array}\right.{\,}\left\{\begin{array}{cc}{x}{}{1}{=}{0}& {}\\ {y}{=}{0}& {}\\ {z}{=}{0}& {}\end{array}\right.\right]$
 (5) 
>

$\mathrm{map}\left(\mathrm{IntersectionMultiplicity}\,\mathrm{dec}\,F\,R\right)$

$\left[\left[\left[{1}{\,}{\mathrm{regular\_chain}}\right]\right]{\,}\left[\left[{2}{\,}{\mathrm{regular\_chain}}\right]\right]{\,}\left[\left[{2}{\,}{\mathrm{regular\_chain}}\right]\right]{\,}\left[\left[{2}{\,}{\mathrm{regular\_chain}}\right]\right]\right]$
 (6) 


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): 198211.


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): 4560.



Compatibility


•

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



