Projections of tropical varieties: Tropical bases and self-intersection points

From CRCG-Wiki

Given an ideal $I=\langle f_1,\ldots, f_r \rangle\in K[x_1,\ldots,x_n]$, where $K$ is a valuated field, we study projections $\pi:R^n\to R^{dim(I)}$ and the image $\pi(T(I))$. We show that we can construct small tropical bases of $I$. Furthermore we try to analyze the combinatorial structure of the image $\pi(T(I))$, i.e. how many self-intersection points can a tropical curve have? (Based on joint work with Thorsten Theobald)