This definition is a discriminant for a singular point on a scalar valued multilinear map.
Cayley's second hyperdeterminant is defined for a restricted range of hypermatrix formats (including the hypercubes of any dimensions).
The quartic expression for the Cayley's hyperdeterminant of hypermatrix A with components aijk, i, j, k ∊ {0, 1} is given by This expression acts as a discriminant in the sense that it is zero if and only if there is a non-zero solution in six unknowns xi, yi, zi, (with superscript i = 0 or 1) of the following system of equations The hyperdeterminant can be written in a more compact form using the Einstein convention for summing over indices and the Levi-Civita symbol which is an alternating tensor density with components εij specified by ε00 = ε11 = 0, ε01 = −ε10 = 1: Using the same conventions we can define a multilinear form Then the hyperdeterminant is zero if and only if there is a non-trivial point where all partial derivatives of f vanish.
In the general case a hyperdeterminant is defined as a discriminant for a multilinear map f from finite-dimensional vector spaces Vi to their underlying field K which may be
Each format number must be greater than or equal to the other, therefore only square matrices S have hyperdeterminants and they can be identified with the determinant det(S).
Applying the definition of the hyperdeterminant as a discriminant to this case requires that det(S) is zero when there are vectors X and Y such that the matrix equations SX = 0 and YS = 0 have solutions for non-zero X and Y.
Since the hyperdeterminant is homogeneous in its variables it has a well-defined degree that is a function of the format and is written N(k1, ..., kr).
For example, a hyperdeterminant is said to be of boundary format when the largest format number is the sum of the others and in this case we have[6] For hyperdeterminants of dimensions 2r, a convenient generating formula for the degrees Nr is[7] In particular for r = 2,3,4,5,6 the degree is respectively 2, 4, 24, 128, 880 and then grows very rapidly.
The full domain of cases in which the product rule can be generalized is still a subject of research.
Using the multiplication rule above on the hyperdeterminant of a hypermatrix H times a matrix S with determinant equal to one gives In other words, the hyperdeterminant is an algebraic invariant under the action of the special linear group SL(n) on the hypermatrix.
The transformation can be equally well applied to any of the vector spaces on which the multilinear map acts to give another distinct invariance.
In other words, for a given hypermatrix format, all the polynomial algebraic invariants with integer coefficients can be formed using addition, subtraction and multiplication starting from a finite number of them.
In the case of a 2 × 2 × 2 hypermatrix, all such invariants can be generated in this way from Cayley's second hyperdeterminant alone, but this is not a typical result for other formats.
[9] The second hyperdeterminant was invented and named by Arthur Cayley in 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".
[10] 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.
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.
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.