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(k_{1},…,k_{n}) for a hyperdeterminant of size (k_{1}+1) x … x (k_{n}+1) which is

Σ N(k_{1},…,k_{n}) z_{1}^{k1} … z_{n}^{kn} = 1/(1 – Σ (i-1)e_{i}(z_{1},…,z_{n}))^{2}

Where e_{i}(z_{1},…,z_{n}) 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 N_{k} z^{k}/k! = e^{-2x}/(1-x)^{2}

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

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This entry was posted on October 18, 2008 at 5:24 pm and is filed under Uncategorized.

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