Singularities of diagonals of Laurent series for rational functions

TL;DR

This paper investigates the analytic continuation and singularities of the diagonal of Laurent series expansions of rational functions in multiple variables, identifying the Landau variety as key to understanding their singularities.

Contribution

It establishes conditions under which the diagonal can be analytically continued and characterizes its singularities via the Landau variety related to the Newton polyhedron.

Findings

Diagonal can be analytically continued outside the Landau variety.

Landau variety is constructed from discriminants of truncations of the denominator.

Complete description of singularities for diagonals of rational functions.

Abstract

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.