  A 'prehomogeneous vector space' is a rational representation
$\rho:G\to\mathrm{GL}(V)$ of a connected complex linear algebraic group $G$
that has a Zariski open orbit $\Omega\subset V$. Mikio Sato showed that the
hypersurface components of $D:=V\setminus \Omega$ are related to the rational
characters $H\to\mathrm{GL}(\mathbb{C})$ of $H$, an algebraic abelian quotient
of $G$. Mimicking this work, we investigate the 'additive functions' of $H$,
the homomorphisms $\Phi:H\to (\mathbb{C},+)$. Each such $\Phi$ is related to an
'additive relative invariant', a rational function $h$ on $V$ such that $h\circ
\rho(g)-h=\Phi(g)$ on $\Omega$ for all $g\in G$. Such an $h$ is homogeneous of
degree $0$, and helps describe the behavior of certain subsets of $D$ under the
$G$--action.
  For those prehomogeneous vector spaces with $D$ a type of hypersurface called
a linear free divisor, we prove there are no nontrivial additive functions of
$H$, and hence $H$ is an algebraic torus. From this we gain insight into the
structure of such representations and prove that the number of irreducible
components of $D$ equals the dimension of the abelianization of $G$. For some
special cases ($G$ abelian, reductive, or solvable, or $D$ irreducible) we
simplify proofs of existing results. We also examine the homotopy groups of
$V\setminus D$.
