\lambda, \dots, a_n + \lambda).$$

We define $\operatorname{Log}\colon K^n \to {\mathbf}{TP}^n$ by $$\operatorname{Log}[x_0:\ldots:x_n] = [\log|x_0|_K:\ldots:\log|x_n|_K]$$ where by convention, $\log 0 = -\infty$.

When $X$ is a projective variety over $K$ that intersects the torus, $(K^\times)^n$, its *tropicalization* is the closure (in the Euclidean topology) of the image of $X \cap K^n$ under $\operatorname{Log}$. We denote the tropicalization of $X$ by $\operatorname{Trop}(X)$.

For varieties embedded in a product of projective spaces, there is a similarly defined $\operatorname{Log}$ map and tropicalization inside a corresponding product of tropical projective spaces. For tropicalization of varieties embedded into general toric varieties, we refer the reader to Payne’s article [@Payne]. In this paper, we talk about tropicalizations of varieties embedded into toric varieties, but it will always be sufficient to think about products of projective spaces.

Metric graphs {#sec:metric-graphs}
-------------

Let $\Gamma$ be a topological space with a distance function $\operatorname{d}\colon \Gamma \times \Gamma \to {\mathbf}R \cup \{\infty\}$. We call $\Gamma$ a *metric graph* if it admits a 1-dimensional simplicial structure where every edge $e$ (aka 1-simplex), with the induced distance function $\operatorname{d}_e$ is isometric to a closed interval: $[0,l] \subseteq {\mathbf}R \cup \{\infty\}$. We allow the possibility of infinite edges—isometric to $[0,\infty]$— but we require 