# Algorithmic Aspects of Gröbner Fans and Tropical Varieties

By Anders Nedergaard Jensen
PhD Dissertations
August 2007
Abstract:
The Gröbner fan of a polynomial ideal $I\subseteq k[x_1,\dots,x_n]$ is a polyhedral complex in $\mathbb{R}^n$ whose maximal cones are in bijection with the reduced Gröbner bases of $I$. In tropical algebraic geometry the tropical variety of an ideal is defined. It is the image of an algebraic variety over the Puiseux series field under the negative valuation map. Another description of it is as a certain subcomplex of the Gröbner fan. In this dissertation we study the structure of both polyhedral fans and suggest algorithms for computing them.