# sylvesterMatrix -- Sylvester-type matrix for the hyperdeterminant of a matrix of boundary shape

## Synopsis

• Usage:
sylvesterMatrix M
• Inputs:
• M, , an $n$-dimensional matrix of boundary shape $(k_1+1)\times\cdots\times (k_n+1)$ (that is, $2 max\{k_1,\ldots,k_n\} = k_1+\ldots+k_n$).
• Outputs:
• , a particular square matrix whose determinant is $det(M)$ (up to sign), introduced by Gelfand, Kapranov, and Zelevinsky.

## Description

This is an implementation of Theorem 3.3, Chapter 14, in Discriminants, Resultants, and Multidimensional Determinants.

 i1 : M = randomMultidimensionalMatrix {4,2,3} o1 = {{{8, 1, 3}, {7, 8, 3}}, {{3, 7, 8}, {8, 5, 7}}, {{8, 5, 2}, {3, 6, 3}}, ------------------------------------------------------------------------ {{6, 8, 6}, {9, 3, 7}}} o1 : 3-dimensional matrix of shape 4 x 2 x 3 over ZZ i2 : S = sylvesterMatrix M o2 = | 8 1 3 0 0 0 7 8 3 0 0 0 | | 0 8 0 1 3 0 0 7 0 8 3 0 | | 0 0 8 0 1 3 0 0 7 0 8 3 | | 3 7 8 0 0 0 8 5 7 0 0 0 | | 0 3 0 7 8 0 0 8 0 5 7 0 | | 0 0 3 0 7 8 0 0 8 0 5 7 | | 8 5 2 0 0 0 3 6 3 0 0 0 | | 0 8 0 5 2 0 0 3 0 6 3 0 | | 0 0 8 0 5 2 0 0 3 0 6 3 | | 6 8 6 0 0 0 9 3 7 0 0 0 | | 0 6 0 8 6 0 0 9 0 3 7 0 | | 0 0 6 0 8 6 0 0 9 0 3 7 | 12 12 o2 : Matrix ZZ <--- ZZ i3 : det M o3 = 910015877 i4 : det S o4 = 910015877 i5 : assert(oo == ooo or oo == -ooo)