# A Gr\"obner basis for the graph of the reciprocal plane

## TL;DR

This paper establishes a connection between the Hilbert series of a certain algebraic variety related to hyperplane arrangements and matroid theory, providing a Gr"obner basis proof of their equivalence.

## Contribution

It introduces an extended no broken circuit complex for matroids and uses it to prove the equivalence of two Hilbert series expressions via Gr"obner basis methods.

## Key findings

- The Hilbert series expressions from Orlik-Terao and Huh-Katz agree.
- A new extended no broken circuit complex is defined for matroids.
- A Gr"obner basis argument confirms the equivalence of the two series.

## Abstract

Given the complement of a hyperplane arrangement, let $\Gamma$ be the closure of the graph of the map inverting each of its defining linear forms. The characteristic polynomial manifests itself in the Hilbert series of $\Gamma$ in two different-seeming ways, one due to Orlik and Terao and the other to Huh and Katz. We define an extension of the no broken circuit complex of a matroid and use it to give a direct Gr\"obner basis argument that the polynomials extracted from the Hilbert series in these two ways agree.