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: