Asian Journal of Mathematics

Volume 20 (2016)

Number 3

Functions dividing their Hessian determinants and affine spheres

Pages: 503 – 530



Daniel J. F. Fox (Departamento de Matemáticas del Área Industrial, Escuela Técnica Superior de Ingeniería y Diseño Industrial, Universidad Politécnica de Madrid, Spain)


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.


affine sphere, prehomogeneous vector space, hyperdeterminant, Monge–Ampère equation

2010 Mathematics Subject Classification

Primary 11S90, 15A15, 53A15. Secondary 20G05.

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