Mathematics – Algebraic Geometry
Scientific paper
2007-05-16
Collect. Math. 59 (2008), no. 2, 129--165
Mathematics
Algebraic Geometry
33 pages
Scientific paper
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 Puiseux-valued ``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 Singular and Gfan if the base field are the rational numbers. As a byproduct we get an algorithm to compute the Puiseux expansion of a space curve singularity in (K^{n+1},0).
Jensen Anders Nedergaard
Markwig Hannah
Markwig Thomas
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