Collectanea mathematica

, Volume 59, Issue 2, pp 129–165 | Cite as

An algorithm for lifting points in a tropical variety

  • Anders Nedergaard Jensen
  • Hannah Markwig
  • Thomas Markwig
Article

Abstract

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).

Keywords

Tropical geometry Puiseux series Puiseux parametrisation 

MSC2000

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

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Copyright information

© Universitat de Barcelona 2008

Authors and Affiliations

  • Anders Nedergaard Jensen
    • 1
  • Hannah Markwig
    • 2
  • Thomas Markwig
    • 3
  1. 1.Institut für Mathematik, MA 4-5Technische Universität BerlinBerlinGermany
  2. 2.Institute for Mathematics and its ApplicationsUniversity of MinnesotaMinneapolis
  3. 3.Fachbereich MathematikTechnische Universität KaiserslauternKaiserslauternGermany

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