2018 年 72 巻 2 号 p. 307-331
An identity that relates the Fourier transform of a complex power of homogeneous polynomial functions on a real vector space with a complex power of homogeneous polynomial functions on the dual vector space is called a local functional equation. A rich source of polynomials satisfying local functional equations is the theory of prehomogeneous vector spaces. Almost all known examples of local functional equations are of this type. However, recently, local functional equations of non-prehomogeneous type have been found. In this paper we present new examples of non-prehomogeneous polynomials satisfying a local functional equation. More precisely, we prove a local functional equation for the polarization of an arbitrary homaloidal polynomial, and calculate the associated b-function identities explicitly.
