exceptionalSet -- returns the exceptional set of a matrix

Synopsis

Usage:

exceptionalSet A
Inputs:
- A, a $d\times n$ integer matrix
Outputs:
- a list, a list that indexes the exceptional parameters of $A$

Description

This method takes a $d\times n$ integer matrix $A$ and computes the exceptional parameters of $A$. The exceptional parameters of $A$ are the $\beta\in\CC^d$ such that the rank of the hypergeometric system $H_\beta(A)$ does not take the expected value. The exceptional parameters of $A$ are indexed by a list of pairs $(v,F)$ where $v$ is a vector and $F$ is a list of vectors. The pair $(v,F)$ represents the plane $v+span_\CC F$. The set of exceptional parameters of $A$ is the union of all such planes given by the pairs $(v,F)$.

i1 : A=matrix{{1,1,1,1},{0,1,5,11}}

o1 = | 1 1 1 1  |
     | 0 1 5 11 |

              2       4
o1 : Matrix ZZ  <-- ZZ

i2 : exceptionalSet A

o2 = {{| 2 |, {}}, {| 3 |, {}}, {| 3 |, {}}, {| 4 |, {}}}
       | 4 |        | 4 |        | 9 |        | 9 |

o2 : List

Thus, when $\beta$=(4,9), (3,9), (2,4), or (3,4), the rank of the hypergeometric system $H_\beta(A)$ is higher than expected.

Ways to use exceptionalSet :

exceptionalSet(Matrix)

For the programmer

The object exceptionalSet is a method function.