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TensorComplexes :: hyperdeterminant

hyperdeterminant -- computes the hyperdeterminant of a boundary format tensor

Synopsis

Description

This constructs the hyperdeterminant of a tensor of boundary format, where we say that a $a\times b_1\times \dots \times b_n$ has boundary format if $$ a-\sum_{i=1}^n (b_i-1)=1. $$ We construct the hyperdeterminant as the determinant of a certain square matrix derived from $f$. The hyperdeterminant function outputs the hyperdeterminant itself, whereas the hyperdeterminantMatrix function outputs the matrix used to compute the hyperdeterminant. (For background on computing hyperdeterminants, see Section 14.3 of the book ``Discriminants, resultants, and multidimensional determinants '' by Gelfand-Kapranov-Zelevinsky.)

The following constructs the generic hyperdetermiant of format $3\times 2\times 2$, which is a polynomial of degree 6 consisting of 66 monomials.

i1 : f=flattenedGenericTensor({3,2,2},QQ);

                                4                          3
o1 : Matrix (QQ[x     ..x     ])  <--- (QQ[x     ..x     ])
                 0,0,0   2,1,1              0,0,0   2,1,1
i2 : S=ring f;
i3 : h=hyperdeterminant f;
i4 : degree h

o4 = {6}

o4 : List
i5 : #terms h

o5 = 66

Caveat

There is bug involving the graded structure of the output. Namely, the code assumes that all entries of f have degree 1, and gives the wrong graded structure if this is not the case. If ring f is not graded, then the code gives an error.

See also

Ways to use hyperdeterminant :

For the programmer

The object hyperdeterminant is a method function.