# The tropical Cayley-Menger variety

## TL;DR

This paper explores the tropicalization of the Cayley-Menger variety, revealing its structure as a Minkowski sum of ultrametrics and providing a new tropical proof of Laman's theorem, advancing algebraic rigidity theory.

## Contribution

It characterizes the tropical Cayley-Menger variety for d=2 as a Minkowski sum of ultrametrics and offers a novel tropical proof of Laman's theorem.

## Key findings

- Tropical Cayley-Menger variety equals Minkowski sum of ultrametrics for d=2
- Polyhedral structure of the tropical variety is described
- Provides a new tropical proof of Laman's theorem

## Abstract

The Cayley-Menger variety is the Zariski closure of the set of vectors specifying the pairwise squared distances between $n$ points in $\mathbb{R}^d$. This variety is fundamental to algebraic approaches in rigidity theory. We study the tropicalization of the Cayley-Menger variety. In particular, when $d = 2$, we show that it is the Minkowski sum of the set of ultrametrics on $n$ leaves with itself, and we describe its polyhedral structure. We then give a new, tropical, proof of Laman's theorem.