# Logarithmic concavity for morphisms of matroids

## TL;DR

This paper introduces basis functions for morphisms of matroids, proves their generating functions are strongly log-concave, and generalizes Mason's conjecture using multivariate Tutte polynomials shown to be Lorentzian.

## Contribution

It defines basis for morphisms of matroids, proves strong log-concavity of their generating functions, and extends Mason's conjecture to all morphisms of matroids.

## Key findings

- Basis functions for morphisms of matroids are introduced.
- Generating functions of these bases are strongly log-concave.
- Multivariate Tutte polynomials of morphisms are Lorentzian for small parameters.

## Abstract

Morphisms of matroids are combinatorial abstractions of linear maps and graph homomorphisms. We introduce the notion of basis for morphisms of matroids, and show that its generating function is strongly log-concave. As a consequence, we obtain a generalization of Mason's conjecture on the $f$-vectors of independent subsets of matroids to arbitrary morphisms of matroids. To establish this, we define multivariate Tutte polynomials of morphisms of matroids, and show that they are Lorentzian in the sense of [BH19] for sufficiently small positive parameters.