# Face module for realizable Z-matroids

## TL;DR

This paper introduces the face module for realizable Z-matroids, linking its Hilbert series to the specialized Grothendieck-Tutte polynomial, advancing the algebraic understanding of matroid invariants.

## Contribution

It defines the face module for realizable Z-matroids and connects its Hilbert series to a known polynomial invariant, extending matroid theory.

## Key findings

- Face module for realizable Z-matroids defined
- Hilbert series matches the specialization of Grothendieck-Tutte polynomial
- Provides new algebraic insights into matroid invariants

## Abstract

In this work, we define the face module for a realizable matroid over Z. Its Hilbert series is, indeed, the expected specialization of the Grothendieck - Tutte polynomial defined by Fink and Moci. This work will appear in 'Contributions to Discrete Mathematics'