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 = {{| 4 |, {}}, {| 2 |, {}}, {| 3 |, {}}, {| 3 |, {}}}
| 9 | | 4 | | 4 | | 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.
The object exceptionalSet is a method function.