On the singular structure of graph hypersurfaces (Q719221)

For a graph one can associate its first and second graph polynomials (a.k.a. Shymanzik and Kirchoff polynomials). The vanishing of these polynomials define the graph hypersurfaces. The paper under review studies the singularity loci of graph hypersurfaces. The author considers configuration polynomials, and shows that the multiplicity of a point on a hypersurface defined by a configuration polynomial can be calculated as a corank of a bilinear form. It was known before that the first graph polynomial is a configuration polynomial (cf. first homology of the graph). The author shows that the second graph polynomial is also a configuration polynomial (cf. relative first homology). Hence the author's multiplicity\(=\)corank theorem holds for both polynomials. ``The result indicates that there may be a fruitful interplay between the current research in graph hypersurfaces and Stratified Morse Theory''. As an application the paper gives an explicit description of the tangent cone of the graph hypersurfaces.