  In this article we define an algebraic vertex of a generalized polyhedron and
show that it is the smallest set of points needed to define the polyhedron. We
prove that the indicator function of a generalized polytope $P$ is a linear
combination of indicator functions of simplices whose vertices are algebraic
vertices of $P$. We also show that the indicator function of any generalized
polyhedron is a linear combination, with integer coefficients, of indicator
functions of cones with apices at algebraic vertices and line-cones. The
concept of an algebraic vertex is closely related to the Fourier--Laplace
transform. We show that a point $\mathbf{v}$ is an algebraic vertex of a
generalized polyhedron $P$ if and only if the tangent cone of $P$, at
$\mathbf{v}$, has non-zero Fourier--Laplace transform.
