Configuration polynomials under contact equivalence

TL;DR

This paper explores configuration polynomials, their classification under contact equivalence, and identifies minimal representatives, revealing finiteness of classes up to rank 3 and providing explicit normal forms.

Contribution

It introduces a classification framework for configuration polynomials under contact equivalence and establishes bounds on minimal variable counts, with explicit normal forms for low ranks.

Findings

Number of equivalence classes is finite up to rank 3

Minimal configuration polynomials have at most (r+1 choose 2) variables

Explicit normal forms are provided for ranks 1, 2, and 3

Abstract

Configuration polynomials generalize the classical Kirchhoff polynomial defined by a graph. Their study sheds light on certain polynomials appearing in Feynman integrands. Contact equivalence provides a way to study the associated configuration hypersurface. In the contact equivalence class of any configuration polynomial we identify a polynomial with minimal number of variables; it is a configuration polynomial. This minimal number is bounded by $r+1\choose 2$, where $r$ is the rank of the underlying matroid. We show that the number of equivalence classes is finite exactly up to rank $3$ and list explicit normal forms for these classes.