An introduction to local tropicalization

In this paper we explain four viewpoints on the local tropicalization of formal subgerms of toric germs, which is a local analog of the global tropicalization of subvarieties of algebraic tori. We start by illustrating some of those viewpoints for plane curve singularities, then we pass to arbitrary dimensions. We conclude by describing several variants and extensions of the notion of local tropicalization presented in this paper.