We study the space of binary cubic and quadratic forms over the ring of integers O of an algebraic number field k. By applying the theory of prehomogeneous vector spaces founded by M. Sato and T. Shintani, we can associate the zeta functions for these spaces. Applying these zeta functions, we derive some density theorems on the distributions of discriminants of cubic algebras of O. In the case ...

Let P be the projective space over a field of characteristic zero. If F is a homogeneous polynomial we say that F is homaloidal if the polar map ∂F defined by the partial derivatives of F is a birational selfmap of P. Although the problem of determining homaloidal polynomials has a classical flavour the theme only recently was raised in an algebro-geometric context by Dolgachev ([Do]), followin...

Let Q be a quiver with dimension vector α prehomogeneous under the action of product general linear groups GL(α) on representation variety Rep(Q,α). We study geometric properties zero sets semi-invariants this space. It is known that for large numbers N, nullcone in Rep(Q,N⋅α) becomes complete intersection. First, we show it also reduced. Then, using Bernstein–Sato polynomials, discuss some cri...

Abstract Let X be a real prehomogeneous vector space under reductive group G , such that is an absolutely spherical -variety with affine open orbit. We define local zeta integrals involve the integration of Schwartz–Bruhat functions on against generalized matrix coefficients admissible representations $G(\mathbb {R})$ twisted by complex powers relative invariants. establish convergence these in...

In this paper, we consider the structure of rational orbit of some prehomogeneous vector spaces originally motivated by Wright and Yukie. After we give a refinement of WrightYukie’s construction, we apply it to the inner forms of the D5 and E7 types. Parameterizations of reducible algebras for the cubic cases are included.