Singularities of diagonals of Laurent series for rational functions

We study the complete diagonal of the Laurent series expansion of a rational function in $n$-complex variables. For a denominator that is nondegenerate for its Newton polyhedron, we prove that the complete diagonal, initially defined in a logarithmically convex domain, can be analytically continued along any path in the $r$-dimensional complex torus that avoids an explicitly defined complex analytic set $L$ called the Landau variety. This variety is constructed as the union of discriminants associated with specific truncations of the denominator to the faces of its Newton polyhedron.