The multivariate arithmetic Tutte polynomial

TL;DR

This paper introduces an arithmetic multivariate Tutte polynomial, providing new representations, positivity proofs, and geometric interpretations, with applications to arithmetic colorings and flows.

Contribution

It presents an arithmetic version of the multivariate Tutte polynomial and a quasi-polynomial interpolation, along with a generalized Fortuin-Kasteleyn representation.

Findings

Introduces an arithmetic multivariate Tutte polynomial

Provides a new proof of positivity of coefficients

Offers a geometric interpretation in the representable case

Abstract

We introduce an arithmetic version of the multivariate Tutte polynomial, and (for representable arithmetic matroids) a quasi-polynomial that interpolates between the two. A generalized Fortuin-Kasteleyn representation with applications to arithmetic colorings and flows is obtained. We give a new and more general proof of the positivity of the coefficients of the arithmetic Tutte polynomial, and (in the representable case) a geometrical interpretation of them.