Hyperdeterminant - History and Applications

History and Applications

The second hyperdeterminant was invented and named by Arthur Cayley (1845) who was able to write down the expression for the 2×2×2 format, but Cayley went on to use the term for any algebraic invariant and later abandoned the concept in favour of a general theory of polynomial forms which he called "quantics" . For the next 140 years there were few developments in the subject and hyperdeterminants were largely forgotten until they were rediscovered by Gel'fand, Kapranov and Zelevinsky in the 1980s as an offshoot of their work on generalized hypergeometric functions (see preface to GKZ ). This led to them writing their textbook in which the hyperdeterminant is reintroduced as a discriminant. However, GKZ omitted the developments as listed in Pascal (1897) and Lecat (1910, 1911). Indeed, Cayley's first hyperdeterminant is more fundamental than his second, since it is a straightforward generalization the ordinary determinant, and has found recent applications in the Alon-Tarsi conjecture. See Zappa (1997).

Since then the hyperdeterminant has found applications over a wide range of disciplines including algebraic geometry, number theory, quantum computing and string theory.

In algebraic geometry the second hyperdeterminant is studied as a special case of an X-discriminant. A principal result is that there is a correspondence between the vertices of the Newton polytope for hyperdeterminants and the "triangulation" of a cube into simplices (see GKZ)

In quantum computing the invariants on hypermatrices of format 2N are used to study the entanglement of N qubits.

In string theory the hyperdeterminant first surfaced in connection with string dualities and black hole entropy..