Projection of tropical intersection

Let $f1,f2$ be polynomials with generic coeﬃcients on Newton polytopes
$P_1 = conv(e1,e2,e3,e1 + e2)$ and $P_2 = conv(e1,e2,e3,e1 + e3)$ respectively. Compute
$\mathcal{T}(<f1,f2>)$, its projection onto the $xy$-plane, and the dual Newton polytope. Verify
your solution with Macaulay2.

We can read off $\mathcal{T}(<f1,f2>)$ from the Minkowski sum of $P_1$ and $P_2$. It consists of the normal vectors of the mixed cells.
Here is a plot of $P_1 + P_2$ (You have to use Acrobat Reader to see the polytope, the polytope can be rotated):
Minkowski Sum
The two small triangles are non-mixed cells, the other 5 facets are mixed.
We can now try to read off the normals. Alternatively, we can let GFAN do the work:

gfan_tropicalintersection --stable
Q[x,y,z]{x+y+z+x*y, x+y+z+x*z}


Which results in the following output:
RAYS
-1 -1 -1    # 0
0 -1 0    # 1
0 0 -1    # 2
0 1 1    # 3
1 0 0    # 4

MULTIPLICITIES
1    # Dimension 1
1
1
2
1


Projecting onto the xy-plane, we get the following rays:
(-1,-1) Mult: 1
( 0,-1) Mult: 1
( 0,-1) Mult: 2
( 1, 0) Mult: 1

The corresponding Newton polytope looks like this: Unless otherwise stated, the content of this page is licensed under Creative Commons Attribution-ShareAlike 3.0 License