I have a complex curve $P(z,w)=z+w-1=0$. I get the amoeba map $$(z,w)\rightarrow (\log|z|,\log|w|)$$ of this curve. It's look like this http://en.wikipedia.org/wiki/Amoeba_(mathematics) (the first picture).
The Ronkin function is defined by $$R(x,y)=\frac{1}{(2\pi i)^2}\int_{e^x=|z|}\int_{|w|=e^y} \log|1+z+w|\frac{dz}{z}\frac{dw}{w},$$
The gradient of it equals $$\frac{\partial R}{\partial x}= \frac{1}{2\pi i} \int_{|w|=e^y}(\frac{1}{2\pi i} \int_{|z|=e^x} \frac{dz}{z+1+w})\frac{dw}{w}$$ It equals zero if $|z|=e^x>|1+w|$, and $\frac{\alpha}{\pi}$ by the argument principle.
Can anyone explain me that $$(\frac{\partial R(x,y)}{\partial x},\frac{\partial R(x,y)}{\partial y})$$ is map from amoeba to Newton polygone?