  The nonzero level sets of a homogeneous, logarithmically homogeneous, or
translationally homogeneous function are affine spheres if and only if the
Hessian determinant of the function is a multiple of a power or an exponential
of the function. In particular, the nonzero level sets of a homogeneous
polynomial are proper affine spheres if some power of it equals a nonzero
multiple of its Hessian determinant. The relative invariants of real forms of
regular irreducible prehomogeneous vector spaces yield many such polynomials
which are moreover irreducible. For example, the nonzero level sets of the
Cayley hyperdeterminant are affine spheres.
