# Presentations of Transversal Valuated Matroids

## TL;DR

This paper generalizes classical transversal matroid theory to the valuated setting using tropical geometry, explicitly describing the fibers of the tropical Stiefel map and characterizing transversal valuated matroids.

## Contribution

It provides a detailed description of the fibers of the tropical Stiefel map and characterizes when a valuated matroid is transversal, extending classical matroid theorems to the tropical context.

## Key findings

- Explicit description of fibers of the tropical Stiefel map
- Characterization of transversal valuated matroids via initial matroids
- Dual results describing stable intersections and valuated strict gammoids

## Abstract

Given $d$ row vectors of $n$ tropical numbers, $d<n$, the tropical Stiefel map constructs a version of their row space, whose Pl\"ucker coordinates are tropical determinants. We explicitly describe the fibers of this map. From the viewpoint of matroid theory, the tropical Stiefel map defines a generalization of transversal matroids in the valuated context, and our results are the valuated generalizations of theorems of Brualdi and Dinolt, Mason and others on the set of all set families that present a given transversal matroid. We show that a connected valuated matroid is transversal if and only if all of its connected initial matroids are. The duals of our results describe complete stable intersections via valuated strict gammoids.