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# exceptionalSet -- returns the exceptional set of a matrix

- 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.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Quasidegrees.m2:810:0.