Computing tropical varieties

Authors:
T. Bogart;A. N. Jensen;D. Speyer;B. Sturmfels;R. R. Thomas
Affiliations:
Department of Mathematics, University of Washington, Seattle, WA 98195, USA;Institut for Matematiske Fag, Aarhus Universitet, DK-8000 rhus, Denmark;Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA;Department of Mathematics, University of California, Berkeley, CA 94720, USA;Department of Mathematics, University of Washington, Seattle, WA 98195, USA
Venue:
Journal of Symbolic Computation
Year:
2007

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Abstract

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 Grobner fan software 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.