# 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 .