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An algorithm for lifting points in a tropical variety

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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).

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Authors and Affiliations

  1. Institut für Mathematik, MA 4-5, Technische Universität Berlin, 10623, Berlin, Germany

    Anders Nedergaard Jensen

  2. Institute for Mathematics and its Applications, University of Minnesota, 400 Lind Hall 207 Church Street S.E., 55455-0436, Minneapolis, MN

    Hannah Markwig

  3. Fachbereich Mathematik, Technische Universität Kaiserslautern, Postfach 3049, 67653, Kaiserslautern, Germany

    Thomas Markwig

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  1. Anders Nedergaard Jensen
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  2. Hannah Markwig
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  3. Thomas Markwig
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Correspondence to Anders Nedergaard Jensen.

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Jensen, A.N., Markwig, H. & Markwig, T. An algorithm for lifting points in a tropical variety. Collect. Math. 59, 129–165 (2008). https://doi.org/10.1007/BF03191365

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  • DOI: https://doi.org/10.1007/BF03191365

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