The calling sequence IntersectionMultiplicity(P, F, V) returns either the intersection multiplicity of $F$ at $P$ as a nonnegative integer, or $\mathrm{FAIL}$ if the algorithm cannot determine the intersection multiplicity.

The calling sequence IntersectionMultiplicity(P, rc1, R) returns the intersection multiplicity of the regular chain $\mathrm{rc1}$ at $P$.

The calling sequence IntersectionMultiplicity(rc2, F, R) returns a list of pairs, $\left[m\,t\right]$, such that $m$ is either the intersection multiplicity of all points in the vanishing set of the regular chain $t$, or $m=\mathrm{FAIL}$. Moreover, the regular chains, $t$, contained in the output list, form a triangular decomposition of the zero set of $\mathrm{rc2}$.

The system is square. That is, when the first argument is a point and the second is a list, $\#F\=\#V\=\#P$ is required. When the second argument is a regular chain, $P$ must have as many elements as there are variables in $R$. When the first argument is a regular chain, $F$ must have as many equations as there are variables in $R$.

For the calling sequence IntersectionMultiplicity(P, F, V) it is assumed $F$ is a regular sequence in the local ring at $P$.

For the calling sequence IntersectionMultiplicity(rc2, F, R) it is assumed $F$ is a regular sequence in the local ring at all points encoded by $\mathrm{rc2}$.

The local ring at a point $P$ is the set of all polynomial fractions defined at $P$.

Let $f_1\,\dots \,f_d$ be elements in the local ring at some point $P$. We say $f_1\,\dots \,f_d$ is a regular sequence in the local ring at $P$, or a regular sequence for short, when no $f_i$ is zero or a zero-divisor modulo the ideal generated by any subset of the remaining elements of $f_1\,\dots \,f_d$. Moreover, we require $f_1\,\dots \,f_d$ generates a proper ideal in the local ring at $P$.

When the elements of $F$ do not generate a proper ideal, i.e. the ideal generated contains 1, the intersection multiplicity is trivially zero. Hence, the assumption that $F$ is a regular sequence is only required when the intersection multiplicity at $P$ is positive.

method : either fulton, tangentcone, or hybrid (default); specifies the method used to compute intersection multiplicities. This option is ignored when the first argument is a point.

When the first argument is a regular chain, IntersectionMultiplicity calls up to two partial algorithms. Each of these algorithms can be accessed individually using the keyword method, in which case IntersectionMultiplicity would invoke only the algorithm corresponding to the value of the keyword method.

The first partial algorithm, described in [3], uses a generalization of Fulton's bivariate intersection multiplicity algorithm to more than $2$ variables. This algorithm is also the one used by IntersectionMultiplicity when the first argument is a point. When the first argument is a regular chain this algorithm can be invoked directly by specifying method=fulton.

The second partial algorithm, described in [1,2], uses a criterion based on tangent cones to reduce to Fulton's bivariate intersection multiplicity algorithm. This algorithm can be invoked directly using method=tangentcone. Additionally, when method=tangentcone is given and the algorithm does not succeed in computing the intersection multiplicity at every point in $\mathrm{rc2}$, an error is returned rather then a list of pairs.

When method=hybrid, IntersectionMultiplicity first calls the algorithm described in [3], and if it did not succeed in computing the intersection multiplicity at every point in $\mathrm{rc2}$, the algorithm described in [1,2] will be called as well. If neither algorithm succeeds in computing the intersection multiplicity at every point in $\mathrm{rc2}$, the output of the first algorithm is returned, namely, a list of pairs (which may contain $\mathrm{FAIL}$).

For strict backwards compatibility with Maple versions 2021 and earlier, method=tangentcone can be used.

maxshift : nonnegative integer; maximum number of variable reorderings to be applied

The algorithm described in [3] and its generalization for regular chains assign an arbitrary variable ordering to the input. This can be changed manually by giving the variables in $V$ or $R$ in a different order. When this occurs, the corresponding point or regular chain must also be reordered accordingly.

It is possible the algorithm described in [3] fails for one ordering but succeeds for another. Moreover, it is impractical to try all possible variable orderings. Upon detecting a failure, IntersectionMultiplicity will apply a left, circular shift to the variables, when written in decreasing order, up to maxshift many times. The user can also reorder variables manually to work past a failure.

When the first argument is a point, maxshift is set to $1$ by default. When the first argument is a regular chain, maxshift is set to $0$ as default.

maxdepth : nonnegative integer,pos_infinity; the maximum number of recursive calls

When $F$ is not a regular sequence, IntersectionMultiplicity may not always terminate. In this case the maxdepth option can be used to control the maximum allowable amount of recursive calls made by IntersectionMultiplicity.

The default value of maxdepth is proportional to the number of variables times the product of the total degree of all polynomials in $F$. When the number of recursive calls made exceeds maxdepth it is likely that $F$ is not a regular sequence, and hence an error is returned.

When this error occurs there are several ways to proceed. This error suggests it is likely $F$ is not a regular sequence, and hence this constraint should be checked. Alternatively, if $F$ is believed to be a regular sequence, one could try increasing the value of maxdepth.

Setting maxdepth=infinity disables this option, removing the limit on the number of allowable recursive calls. This may be desirable for long computations. When maxdepth is set to $\mathrm{\infty }$, or when maxdepth is much lower than the stacklimit kernel option, an error may occur if the number of recursive calls made exceeds the stacklimit. In this case, if kernelopts(stacklimit) is less than the hard limit set by Maple, one can try increasing kernelopts(stacklimit).

To check whether $F$ is a regular sequence in the local ring at $P$, compute the primary decomposition of each $f_i$ in $F$ and remove any primary components which do not vanish on $P$. If any two $f_i$ share a primary component through $P$, $F$ is not a regular sequence in the local ring at $P$.

When the first argument is a regular chain, this must be repeated for each point encoded in the regular chain.

Both the maxdepth and maxshift options are ignored when method=tangentcone is specified.

The IntersectionMultiplicity command computes the intersection multiplicity of a system of equations at a point or at a group of points encoded by a zero-dimensional regular chain. See the references for a formal definition of the intersection multiplicity.

Unless $\#F\=2$ or the second argument is a regular chain, the underlying algorithms to the IntersectionMultiplicity command may fail, in which case $\mathrm{FAIL}$ is returned.

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](..).

$\mathrm{with}\left(\mathrm{RegularChains}\right)\:$$\mathrm{with}\left(\mathrm{ChainTools}\right)\:$$\mathrm{with}\left(\mathrm{AlgebraicGeometryTools}\right)\:$

$V≔\left[x\,y\,z\,w\right]\:$

$P≔\left[0\,0\,0\,0\right]\:$

$\mathrm{IntersectionMultiplicity}\left(P\,\left[1\,y\,z\,w\right]\,V\right)$

$0$

$\mathrm{IntersectionMultiplicity}\left(P\,\left[x\,y\,z\,w\right]\,V\right)$

$1$

$\mathrm{IntersectionMultiplicity}\left(P\,\left[x^2\,y\,z^3\,w\right]\,V\right)$

$6$

$\mathrm{IntersectionMultiplicity}\left(P\,\left[z{y}^2\,y^5-z^2\,x^5-y^2\,w\right]\,V\right)$

$45$

$\mathrm{IntersectionMultiplicity}\left(P\,\left[xy-z\,x^2{y}^3-z\,x^4-y\,w^2\right]\,V\right)$

$10$

$\mathrm{IntersectionMultiplicity}\left(P\,\left[y^2\,y^3+x\,y^5+z\,y^5+w\right]\,V\right)$

$2$

$\mathrm{IntersectionMultiplicity}\left(P\,\left[x^3\,-x^2+y^2\,z^4-wy\,w^2\right]\,V\right)$

$48$

$Q≔\left[1\,2\,-1\,3\right]\:$

$\mathrm{IntersectionMultiplicity}\left(Q\,\left[{\left(x-1\right)}^3\,{\left(y-2\right)}^2-{\left(x-1\right)}^6\,{\left(z+1\right)}^4-\left(x-1\right)\,w-3\right]\,V\right)$

$24$

$\mathrm{alias}\left(\mathrm{\alpha }=\mathrm{RootOf}\left(u^2+2\right)\right)\:$

$P≔\left[\mathrm{\alpha }\,0\,\mathrm{\alpha }\,0\right]$

$P≔\left[\mathrm{\alpha }\,0\,\mathrm{\alpha }\,0\right]$

$F≔\left[\left(x-\mathrm{\alpha }\right)y-\left(z-\mathrm{\alpha }\right)\,{\left(x-\mathrm{\alpha }\right)}^2{y}^3-\left(z-\mathrm{\alpha }\right)\,{\left(x-\mathrm{\alpha }\right)}^4-y\,w^2\right]$

$F≔\left[\left(x-\mathrm{\alpha }\right)y-z+\mathrm{\alpha }\,{\left(x-\mathrm{\alpha }\right)}^2{y}^3-z+\mathrm{\alpha }\,{\left(x-\mathrm{\alpha }\right)}^4-y\,w^2\right]$

$\mathrm{IntersectionMultiplicity}\left(P\,F\,V\right)$

$10$

$P≔\left[0\,0\,0\,0\right]\:$

$F≔\left[x\,y\,z\,0\right]\:$

$G≔\left[x\,16w\,z\,12w^2\right]\:$

$H≔\left[xy+z\,2w^2\,y^2+z\,x{y}^2\right]\:$

$\mathrm{IntersectionMultiplicity}\left(P\,F\,V\right)$

Error, (in RegularChains:-AlgebraicGeometryTools:-IntersectionMultiplicity) input is not a regular sequence

$\mathrm{IntersectionMultiplicity}\left(P\,G\,V\right)$

Error, (in RegularChains:-AlgebraicGeometryTools:-IntersectionMultiplicity) input is not a regular sequence

$\mathrm{IntersectionMultiplicity}\left(P\,H\,V\right)$

Error, (in RegularChains:-AlgebraicGeometryTools:-IntersectionMultiplicity) input is not a regular sequence

$F≔x^2\left(x+1\right)\:$

$G≔y\left(y+1\right)\:$

$H≔x\left(z+1\right)-y^2\left(x+1\right)-y^2{z}^3\left(y+1\right)\:$

$\mathrm{IntersectionMultiplicity}\left(\left[0\,0\,-1\right]\,\left[F\,G\,H\right]\,\left[x\,y\,z\right]\,\mathrm{maxdepth}=100\right)$

Error, (in RegularChains:-AlgebraicGeometryTools:-IntersectionMultiplicity) maximum recursion depth reached; try using the maxdepth option. See ?IntersectionMultiplicity for details

$\mathrm{with}\left(\mathrm{PolynomialIdeals}\right)\:$

$Q≔\left[\mathrm{PrimaryDecomposition}\left(⟨F\,G⟩\right)\right]$

$Q≔\left[⟨y\,x^2\,x^2\left(x+1\right)\,y\left(y+1\right)⟩\,⟨x^2\,x^2\left(x+1\right)\,y\left(y+1\right)\,y+1⟩\,⟨y\,x^2\left(x+1\right)\,y\left(y+1\right)\,x+1⟩\,⟨x^2\left(x+1\right)\,y\left(y+1\right)\,x+1\,y+1⟩\right]$

$Q≔\mathrm{select}\left(\mathrm{IdealContainment}\,Q\,⟨x\,y\,z+1⟩\right)$

$Q≔\left[⟨y\,x^2\,x^2\left(x+1\right)\,y\left(y+1\right)⟩\right]$

$\mathrm{RadicalMembership}\left(H\,Q\left[\right]\right)$

$\mathrm{true}$

$\mathrm{kernelopts}\left(\mathrm{stacklimit}=50000\right)\:$

$F≔\left[{x\left[1\right]}^4\,{x\left[2\right]}^4-x\left[1\right]\,{x\left[3\right]}^4-x\left[2\right]\,{x\left[4\right]}^4-x\left[3\right]\,{x\left[5\right]}^4-x\left[4\right]\,{x\left[6\right]}^4-x\left[5\right]\,{x\left[7\right]}^4-x\left[6\right]\right]\:$

$P≔\left[0\,0\,0\,0\,0\,0\,0\right]\:$

$V≔\left[x\left[1\right]\,x\left[2\right]\,x\left[3\right]\,x\left[4\right]\,x\left[5\right]\,x\left[6\right]\,x\left[7\right]\right]\:$

$\mathrm{IntersectionMultiplicity}\left(P\,F\,V\,'\mathrm{maxdepth}'=\mathrm{\infty }\right)$

$\mathrm{kernelopts}\left(\mathrm{stacklimit}=100000\right)\:$

$\mathrm{IntersectionMultiplicity}\left(P\,F\,V\,\mathrm{maxdepth}=\mathrm{\infty }\right)$

$16384$

$V≔\left[x\,y\,z\,w\right]\:$

$P≔\left[0\,0\,0\,0\right]\:$

$\mathrm{IntersectionMultiplicity}\left(P\,\left[yz-w\,z^3-w\,y^4-z\,x\right]\,V\right)$

$\mathrm{FAIL}$

$\mathrm{IntersectionMultiplicity}\left(P\,\left[yz-w\,z^3-w\,y^4-z\,x\right]\,\left[w\,x\,y\,z\right]\right)$

$5$

$\mathrm{IntersectionMultiplicity}\left(P\,\left[yz-w\,z^3-w\,y^4-z\,x\right]\,V\,\mathrm{maxshift}=2\right)$

$5$

$P≔\left[0\,0\,0\right]$

$P≔\left[0\,0\,0\right]$

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

$\mathrm{rc}≔\mathrm{Chain}\left(\left[z^5\,y^3+z{y}^2\,x^2+x{y}^4+yz\right]\,R\right)\:$

$\mathrm{IntersectionMultiplicity}\left(P\,\mathrm{rc}\,R\right)$

$30$

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

$\mathrm{rc}≔\mathrm{Chain}\left(\left[w\,z\,y\,x\right]\,R\right)\:$

$\mathrm{out}≔\mathrm{IntersectionMultiplicity}\left(\mathrm{rc}\,\left[x\,y\,z\,1\right]\,R\right)$

$\mathrm{out}≔\left[\left[0\,\mathrm{regular\_chain}\right]\right]$

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

$\left[\left[0\,\left\{\begin{array}{cc}x=0& \\ y=0& \\ z=0& \\ w=0& \end{array}\right.\right]\right]$

$\mathrm{out}≔\mathrm{IntersectionMultiplicity}\left(\mathrm{rc}\,\left[x\,y\,z\,w\right]\,R\right)$

$\mathrm{out}≔\left[\left[1\,\mathrm{regular\_chain}\right]\right]$

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

$\left[\left[1\,\left\{\begin{array}{cc}x=0& \\ y=0& \\ z=0& \\ w=0& \end{array}\right.\right]\right]$

$\mathrm{out}≔\mathrm{IntersectionMultiplicity}\left(\mathrm{rc}\,\left[x^2\,y^3\,z^4\,w\right]\,R\right)$

$\mathrm{out}≔\left[\left[24\,\mathrm{regular\_chain}\right]\right]$

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

$\left[\left[24\,\left\{\begin{array}{cc}x=0& \\ y=0& \\ z=0& \\ w=0& \end{array}\right.\right]\right]$

$\mathrm{out}≔\mathrm{IntersectionMultiplicity}\left(\mathrm{rc}\,\left[x^3\,-x^2+y^2\,z^4-wy\,w^2\right]\,R\right)$

$\mathrm{out}≔\left[\left[48\,\mathrm{regular\_chain}\right]\right]$

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

$\left[\left[48\,\left\{\begin{array}{cc}x=0& \\ y=0& \\ z=0& \\ w=0& \end{array}\right.\right]\right]$

$\mathrm{out}≔\mathrm{IntersectionMultiplicity}\left(\mathrm{rc}\,\left[z{y}^2\,y^5-z^2\,x^5-y^2\,w\right]\,R\right)$

$\mathrm{out}≔\left[\left[45\,\mathrm{regular\_chain}\right]\right]$

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

$\left[\left[45\,\left\{\begin{array}{cc}x=0& \\ y=0& \\ z=0& \\ w=0& \end{array}\right.\right]\right]$

$\mathrm{out}≔\mathrm{IntersectionMultiplicity}\left(\mathrm{rc}\,\left[xy-z\,x^2{y}^3-z\,x^4-y\,w^2\right]\,R\right)$

$\mathrm{out}≔\left[\left[10\,\mathrm{regular\_chain}\right]\right]$

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

$\left[\left[10\,\left\{\begin{array}{cc}x=0& \\ y=0& \\ z=0& \\ w=0& \end{array}\right.\right]\right]$

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

$\mathrm{rc}≔\mathrm{Chain}\left(\left[\left(z^2+2z-1\right)z\,\left(y-z\right)y\,x-z\right]\,R\right)\:$

$F≔\left[\left(x^2+y+z-1\right)z\,x\left(y^2+x+z-1\right)\,\left(z^2+x+y-1\right)y^3\right]\:$

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

$\mathrm{dec}≔\left[\left[0\,\mathrm{regular\_chain}\right]\,\left[1\,\mathrm{regular\_chain}\right]\,\left[3\,\mathrm{regular\_chain}\right]\right]$

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

$\left[\left[0\,\left\{\begin{array}{cc}x-z=0& \\ y=0& \\ z^2+2z-1=0& \end{array}\right.\right]\,\left[1\,\left\{\begin{array}{cc}x-z=0& \\ y-z=0& \\ z^2+2z-1=0& \end{array}\right.\right]\,\left[3\,\left\{\begin{array}{cc}x=0& \\ y=0& \\ z=0& \end{array}\right.\right]\right]$

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

$\mathrm{rc}≔\mathrm{Chain}\left(\left[z\,y\,x\right]\,R\right)\:$

$\mathrm{out}≔\mathrm{IntersectionMultiplicity}\left(\mathrm{rc}\,\left[z{y}^2\,y^5-z^2\,x^5-y^2\right]\,R\,\mathrm{method}='\mathrm{fulton}'\right)$

$\mathrm{out}≔\left[\left[45\,\mathrm{regular\_chain}\right]\right]$

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

$\left[\left[45\,\left\{\begin{array}{cc}x=0& \\ y=0& \\ z=0& \end{array}\right.\right]\right]$

$\mathrm{rc}≔\mathrm{Chain}\left(\left[z-10\,y\,x\right]\,R\right)\:$

$F≔\left[-3y^4{z}^3+6x^4{z}^2-3x^2{y}^2{z}^2-x^2{z}^2+2y^2{z}^2+28x^{2}z+7y^{2}z+z^2-11z+10\,x^{3}z+x{y}^{2}z-xz+x\,-2x^{2}yz-y^{3}z-yz+10y\right]\:$

$\mathrm{IntersectionMultiplicity}\left(\mathrm{rc}\,F\,R\,\mathrm{method}='\mathrm{fulton}'\right)$

$\left[\left[\mathrm{FAIL}\,\mathrm{regular\_chain}\right]\right]$

$\mathrm{IntersectionMultiplicity}\left(\mathrm{rc}\,F\,R\,\mathrm{method}='\mathrm{tangentcone}'\right)$

$\left[\left[3\,\mathrm{regular\_chain}\right]\right]$

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

$\mathrm{rc}≔\mathrm{Chain}\left(\left[w\,z\,y\,x\left(x+1\right)\right]\,R\right)\:$

$\mathrm{out}≔\mathrm{IntersectionMultiplicity}\left(\mathrm{rc}\,\left[xy-z\,y^3-z\,\left(x+1\right)x^4-y\,w\right]\,R\,\mathrm{method}='\mathrm{fulton}'\right)$

$\mathrm{out}≔\left[\left[1\,\mathrm{regular\_chain}\right]\,\left[\mathrm{FAIL}\,\mathrm{regular\_chain}\right]\right]$

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

$\left[\left[1\,\left\{\begin{array}{cc}x+1=0& \\ y=0& \\ z=0& \\ w=0& \end{array}\right.\right]\,\left[\mathrm{FAIL}\,\left\{\begin{array}{cc}x=0& \\ y=0& \\ z=0& \\ w=0& \end{array}\right.\right]\right]$

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

$\mathrm{rc}≔\mathrm{Chain}\left(\left[z\,y\,x\right]\,R\right)\:$

$\mathrm{out}≔\mathrm{IntersectionMultiplicity}\left(\mathrm{rc}\,\left[z{y}^2\,y^5-z^2\,x^5-y^2\right]\,R\,\mathrm{method}='\mathrm{tangentcone}'\right)$

Error, (in RegularChains:-AlgebraicGeometryTools:-IntersectionMultiplicity) h top is singular

[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.

The method=tangentcone corresponds to the algorithm in Maple 2020 and 2021.

The RegularChains:-AlgebraicGeometryTools:-IntersectionMultiplicity command was introduced in Maple 2020.

For more information on Maple 2020 changes, see Updates in Maple 2020.

The RegularChains:-AlgebraicGeometryTools:-IntersectionMultiplicity command was updated in Maple 2022.

The P parameter was introduced in Maple 2022.

The method, maxshift and maxdepth options were introduced in Maple 2022.

For more information on Maple 2022 changes, see Updates in Maple 2022.