polarDegrees -- Computes the polar degrees of a projective toric variety

Synopsis

• Usage:

  polarDegrees(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 List, this can be set to HashTable to return a HashTable with all computed values
  • TextOutput (missing documentation) => ..., default value "Quiet",
• Outputs:
  • pd, a list, the polar degrees of the projective toric variety X_A.

Description

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 : polarDegrees(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 = {53, 85, 35}

o2 : List

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 : pdh=polarDegrees(A,Output=>HashTable);

i5 : pdh#"polar degrees"

o5 = {45, 98, 81, 28}

o5 : List

i6 : pdh#"dual degree"

o6 = 45

o6 : QQ

i7 : pdh#"dual codim"

o7 = 1

i8 : pdh#"ED"

o8 = 252

o8 : QQ

i9 : pdh#"degree"

o9 = 28

o9 : QQ