sylvesterMatrix(MultidimensionalMatrix) -- Sylvester-type matrix for the hyperdeterminant of a matrix of boundary shape

Synopsis

Function: sylvesterMatrix
Usage:

sylvesterMatrix M
Inputs:
- M, a multidimensional matrix, 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 matrix, 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)

Ways to use this method: