Singular structure of Feynman diagrams

It is shown that the singularities of any Feynman diagram $G_k(x_1,\dots,x_k)$ in the coordinate space lie on an algebraic surface. For diagrams with one internal vertex, the equation of this surface has the form $\det S=0$, where $S$ is the matrix composed of the elements $s_{jj'}=(x_j-x_j')^2$. In the general case, the equation of the singularity surface is obtained as the necessary and sufficient condition for the existence of a nontrivial solution to a homogeneous algebraic system of equations, this system being derived by means of the concept of the wave front of a generalized function. It is shown how this system of equations can be obtained from the ordinary $\alpha$ representation for Feynmml diagrams.