On the Singular Structure of Graph Hypersurfaces

TL;DR

This paper establishes a connection between the singular loci of graph hypersurfaces and their rank loci, revealing a linear algebra-based characterization that links graph polynomials to configuration polynomials.

Contribution

It proves that the singular loci of graph hypersurfaces correspond to their rank loci and shows the second graph polynomial is a configuration polynomial, bridging graph hypersurfaces and stratified Morse theory.

Findings

Singular loci correspond to rank loci in graph hypersurfaces

Second graph polynomial is a configuration polynomial

Potential for new insights via Stratified Morse Theory

Abstract

We show that the singular loci of graph hypersurfaces correspond set-theoretically to their rank loci. The proof holds for all configuration hypersurfaces and depends only on linear algebra. To make the conclusion for the second graph hypersurface, we prove that the second graph polynomial is a configuration polynomial. The result indicates that there may be a fruitful interplay between the current research in graph hypersurfaces and Stratified Morse Theory.