Computing Tropical Varieties

Details Computing Tropical Varieties Computing Tropical Varieties

2005-07-27

arxiv.org/abs/math/0507563v1

J. Symb. Comput. 42 (2007), no. 1-2, 54--73

Mathematics

Algebraic Geometry

22 pages, 2 figures

Scientific paper

10.1016/j.jsc.2006.02.004

The tropical variety of a $d$-dimensional prime ideal in a polynomial ring with complex coefficients is a pure $d$-dimensional polyhedral fan. This fan is shown to be connected in codimension one. We present algorithmic tools for computing the tropical variety, and we discuss our implementation of these tools in the Gr\"obner fan software \texttt{Gfan}. Every ideal is shown to have a finite tropical basis, and a sharp lower bound is given for the size of a tropical basis for an ideal of linear forms.

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