edDeg -- Computes the (generic) Euclidean distance degree of a projective toric variety

Synopsis

Usage:

edDeg(A)
Inputs:
- A, a matrix, a full rank integer matrix with the vector (1,1,...,1) in its row space defining a projective toric variety X_A
Optional inputs:
- ForceAmat (missing documentation) => a Boolean value, default value false, if A defines a codimension two toric variety a faster method will be used by default, setting this to true forces the general purpose method
- Output (missing documentation) => a list, default value ZZ, this can be set to HashTable to return a HashTable with all computed values
- TextOutput (missing documentation) => ..., default value "Quiet",
Outputs:
- ED, an integer, the (generic) Euclidean distance degree of the projective toric variety X_A.

Description

This function computes (generic) Euclidean distance degree the projective toric variety X_A, we do not assume that X_A is normal. The default output is a list of polar degrees; other values of interest computed by the program are also output. To suppress text output use the option Output =>HashTable. This function uses polarDegrees internally.

i1 : A=matrix{{0, 0, 0, 1, 1,5}, {7,0, 1, 3, 0, -2},{1,1, 1, 1, 1, 1}}

o1 = | 0 0 0 1 1 5  |
     | 7 0 1 3 0 -2 |
     | 1 1 1 1 1 1  |

              3       6
o1 : Matrix ZZ  <-- ZZ

i2 : edDeg(A)

The toric variety has degree = 35
The dual variety has degree = 53, and codimension = 1
Chern-Mather Volumes: (V_0,..,V_(d-1)) = {-12, 20, 35}
Polar Degrees: {53, 85, 35}
ED Degree = 173

                         5      4      3
Chern-Mather Class: - 12h  + 20h  + 35h

o2 = 173

o2 : QQ

i3 : A=matrix{{3, 0, 0, 1, 1,2}, {3,5,0,2,1,3},{0, 1, 2, 0, 2,0},{1, 1, 1, 1, 1,1}}

o3 = | 3 0 0 1 1 2 |
     | 3 5 0 2 1 3 |
     | 0 1 2 0 2 0 |
     | 1 1 1 1 1 1 |

              4       6
o3 : Matrix ZZ  <-- ZZ

i4 : time edDeg(A)

The toric variety has degree = 28
The dual variety has degree = 45, and codimension = 1
Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
Polar Degrees: {45, 98, 81, 28}
ED Degree = 252

                       5      4      3      2
Chern-Mather Class: 20h  + 23h  + 31h  + 28h
     -- used 0.568285 seconds

o4 = 252

o4 : QQ

i5 : time edDeg(A,ForceAmat=>true)

The toric variety has degree = 28
The dual variety has degree = 45, and codimension = 1
Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
Polar Degrees: {45, 98, 81, 28}
ED Degree = 252

                       5      4      3      2
Chern-Mather Class: 20h  + 23h  + 31h  + 28h
     -- used 1.86344 seconds

o5 = 252

o5 : QQ

Ways to use edDeg :

edDeg(Matrix)

For the programmer

The object edDeg is a method function with options.