EDPolytopeCD2: A Package to Compute the Euclidean Distance Degree and Polar Degrees of A Toric Variety with Enhancements in Codimension 2

This page is supplementary material for the article Polar Degrees and Closest Points in Codimension Two by Martin Helmer and Bernt Ivar Utstøl Nødland. It describes Version 2.5 of a Macaulay2 package to compute the Euclidean distance degree, polar degrees and Chern-Mather class of a projective toric variety combinitorially with speed enhancments in the codimension two case. A download link and examples of use are given below. The package is implemented in Macaulay2. The package is called "EDPolytopeCD2.m2" and may be loaded with the command: needsPackage "EDPolytopeCD2".

A version of this package will be available in future releases of Macaulay2 under the name ToricInvariants.

Description

Let

A

be a

d ⨯ n

integer matrix of rank

d

which has the length

n

vector

(1,1,1,...,1)

in its row space and let

X_{A}

be the

(d-1)

-dimensional toric variety in

P^{n-1}

defined by the polytope

Conv(A)

The EDPolytope package gives methods to compute:

The Euclidean distance degree of $X_{A}$ , $EDdeg(X_{A})$ .
The polar degrees of $X_{A}$ , $(δ_{0} (X_{A}),..., δ_{d-1} (X_{A}))$ .
The Chern-Mather class of $X_{A}$ and the Chern-Mather volumes of the polytope $Conv(A)$ .

Methods

The first three methods are for codimension two toric varieties only. These methods use the Gale dual matrix

B

A

internally (as described in Polar Degrees and Closest Points in Codimension Two) and will provide increased performance vs. the general purpose methods below.

EDdegCD2:
- Input: An integer $(n-2) ⨯ n$ matrix $A$ of rank $n-2$ with $(1,1,1,...,1)$ in its row space so that $X_{A}$ is a codimension two projective toric variety.
- Output: The integer $EDdeg(X_{A})$ . Additionally text output is displayed showing both the sums of the Chern-Mather volumes of all faces of each dimension in the corresponding polytope $Conv(A)$ (starting from dimension zero) and the list of polar degrees.
PolarDegreesCD2:
- Input: An integer $(n-2) ⨯ n$ matrix $A$ of rank $n-2$ with $(1,1,1,...,1)$ in its row space so that $X_{A}$ is a codimension two projective toric variety.
- Output: A list containing the polar degrees $(δ_{0} (X_{A}),..., δ_{d-1} (X_{A}))$ . Additionally text output is displayed showing both the sums of the Chern-mather volumes of all faces of each dimension in the corresponding polytope $Conv(A)$ (starting from dimension zero) and the ED degree of the corresponding toric variety $X_{A}$ .
CMClassCD2:
- Input: An integer $(n-2) ⨯ n$ matrix $A$ of rank $n-2$ with $(1,1,1,...,1)$ in its row space so that $X_{A}$ is a codimension two projective toric variety.
- Output: The Chern-Mather class, $c_{M} (X_{A})$ , in the Chow ring of $P^{n-1}$ . We use the symbol $h$ to represent the rational equivalence class of a hyperplane in the Chow ring. Additionally text output is displayed showing $EDdeg(X_{A})$ and the polar degrees.
CMVolumesCD2:
- Input: An integer $(n-2) ⨯ n$ matrix $A$ of rank $n-2$ with $(1,1,1,...,1)$ in its row space so that $X_{A}$ is a codimension two projective toric variety.
- Output: A list of the sums of the Chern-Mather volumes of all faces of each codimension in the corresponding polytope $Conv(A)$ , starting with codimension zero.

The methods below are the general purpose methods implemented in the first EDPolytope package based on the results from the article Nearest Points on Toric Varieties .

EDdeg:
- Input: An integer $d ⨯ n$ matrix $A$ of rank $d$ with $(1,1,1,...,1)$ in its row space.
- Output: The integer $EDdeg(X_{A})$ . Additionally text output is displayed showing both the sums of the Chern-Mather volumes of all faces of each dimension in the corresponding polytope $Conv(A)$ (starting from dimension zero) and the list of polar degrees.
PolarDegrees:
- Input: An integer $d ⨯ n$ matrix $A$ of rank $d$ with $(1,1,1,...,1)$ in its row space.
- Output: A list containing the polar degrees $(δ_{0} (X_{A}),..., δ_{d-1} (X_{A}))$ . Additionally text output is displayed showing both the sums of the Chern-mather volumes of all faces of each dimension in the corresponding polytope $Conv(A)$ (starting from dimension zero) and the ED degree of the corresponding toric variety $X_{A}$ .
CMClass:
- Input: An integer $d ⨯ n$ matrix $A$ of rank $d$ with $(1,1,1,...,1)$ in its row space.
- Output: The Chern-Mather class, $c_{M} (X_{A})$ , in the Chow ring of $P^{n-1}$ . We use the symbol $h$ to represent the rational equivalence class of a hyperplane in the Chow ring. Additionally text output is displayed showing $EDdeg(X_{A})$ and the polar degrees.
CMVolumes:
- Input: An integer $d ⨯ n$ matrix $A$ of rank $d$ with $(1,1,1,...,1)$ in its row space.
- Output: A list of the sums of the Chern-Mather volumes of all faces of each codimension in the corresponding polytope $Conv(A)$ , starting with codimension zero.

For use by programmers; in all methods above, except CMVolumes and CMVolumesCD2, the user may optionally ask for the output in the form of a HashTable containing all computed information including $EDdeg(X_{A})$ , the polar degrees of $X_{A}$ , the degree of $X_{A}$ , the degree of the dual variety, the list of Chern-Mather volumes (starting from dimension zero), and the Chern-Mather class of $X_{A}$ . This option is accessed by specifying Output=>HashTable, see the example below. Additional text output may also be requested using the option TextOutput=>"All".

Examples

To apply the EDPolytope package to Example 2.1 from Polar Degrees and Closest Points in Codimension Two we would enter the following:


needsPackage "EDPolytopeCD2";
A=matrix{{-2,-2,1,0,0},{4,0,0,1,0},{1,1,1,1,1}}  ;
pds=PolarDegreesCD2(A);
ed=EDdegCD2(A);
cm=CMClassCD2(A);

For examples of the general purpose (i.e. any codimension) methods "EDdeg", "PolarDegrees", "CMClass", and "CMVolumes" see the documentation for version 2.2 of the EDPolytope package here: EDPolytope package.

Download

The package file "EDPolytopeCD2.m2" may be downloaded below:

The Article

Martin Helmer, Bernt Ivar Utstøl Nødland. Polar Degrees and Closest Points in Codimension Two