Tropical Varieties And Gfan

Compute the following. You may use the software Gfan.

1. Find a universal Groebner basis for the ideal $I = \langle ac-b^2, bd-c^2\rangle \subset \mathbb{Q}[a,b,c,d]$. How many cones are there of each dimension in the Groebner fan of $I$? Find $\mathcal{T}(I)$ by hand; then check your answer with Gfan.
2. How many edges are there in the $3$-dimensional permutahedron? How would you use Gfan to answer this?
3. Let $f_1 = x+y+z$ and $f_2=x^2+y^2+xz+yz$. Is $\mathcal{T}(f_1) \cap \mathcal{T}(f_2) = \mathcal{T}(\langle f_1, f_2\rangle)$? First try to answer without a computer, then verify your answer with Gfan.

Hint: Gfan is installed on the servers (ima-aga-…). Read the Gfan manual for instructions and examples. Look up the apps: gfan _bases, _topolyhedralfan, _tropicalhypersurface, _tropicalintersection, _tropicalstartingcone, _tropicaltraverse, _tropicalbruteforce, …