## The Degree of Hyperdeterminants

October 18, 2008

Working out hyperdeterminants is an involved process. Apart from the smallest ones they have huge numbers of terms. There is a bit more hope in the task of working out the degree of the hyperdeterminant of a given format. We have already seen that Cayley’s hyperdeterminant for a 2x2x2 hypermatrix is of degree 4 and the hyperdeterminant of size 2x2x2x2 is of degree 24. Of course we also know that determinants of size N x N have degree N.

For the general case there is in fact a generating formula for the degree N(k1,…,kn) for a hyperdeterminant of size (k1+1) x … x (kn+1) which is

Σ N(k1,…,kn) z1k1 … znkn = 1/(1 – Σ (i-1)ei(z1,…,zn))2

Where ei(z1,…,zn) is the ith elementary symmetric polynomial in n variables. For a derivation you should look at the hyperdeterminant book.

For the case of a hyperdeterminant of size 2n you can use this simpler generating function

Σ Nk zk/k! = e-2x/(1-x)2

Which gives us the sequence for Nk  =2, 4, 24, 128, 880, 6816, 60032, 589312, 6384384, 75630080, 972387328, etc . This is also known as sequence A087981.