Functions dividing their Hessian determinants and affine spheres

TL;DR

This paper characterizes when the level sets of certain homogeneous functions form affine spheres, linking their Hessian determinants to specific functional forms and providing examples from prehomogeneous vector spaces.

Contribution

It establishes a precise criterion for functions whose level sets are affine spheres based on their Hessian determinants and explores examples from algebraic structures.

Findings

Level sets of specific homogeneous functions are affine spheres if their Hessian determinants follow particular forms.

Nonzero level sets of homogeneous polynomials are proper affine spheres under certain conditions.

Examples include level sets of the Cayley hyperdeterminant as affine spheres.

Abstract

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.