An algorithm for lifting points in a tropical variety

The aim of this paper is to give a constructive proof of one of the basic theorems of tropical geometry: given a point on a tropical variety (defined using initial ideals), there exists a Puiseuxvalued ``lift'' of this point in the algebraic variety.This theorem is so fundamental because it justifies why a tropical variety (defined combinatorially using initial ideals) carries information about algebraic varieties: it is the image of an algebraic variety over the Puiseux series under the valuation map. We have implemented the “lifting algorithm” using and Gfan if the base field is $\mathbb{Q}$. As a byproduct we get an algorithm to compute the Puiseux expansion of a space curve singularity in $(K^{n+1}, 0)$.