Collectanea mathematica

, Volume 59, Issue 2, pp 129-165

First online:

An algorithm for lifting points in a tropical variety

  • Anders Nedergaard JensenAffiliated withInstitut für Mathematik, MA 4-5, Technische Universität Berlin Email author 
  • , Hannah MarkwigAffiliated withInstitute for Mathematics and its Applications, University of Minnesota
  • , Thomas MarkwigAffiliated withFachbereich Mathematik, Technische Universität Kaiserslautern

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access


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” usingSingular and Gfan if the base field is ℚ. As a byproduct we get an algorithm to compute the Puiseux expansion of a space curve singularity in (K n+1, 0).


Tropical geometry Puiseux series Puiseux parametrisation


Primary 13P10, 51M20, 16W60, 12J25 Secondary 14Q99, 14R99