Newton Polygons With No Common Slopes

Let $f_1, f_2 \in \mathbb{C}[x,y]$ be polynomials with Newton polygons $P$ and $Q$ respectively. Suppose edges of $P$ and $Q$ have distinct slopes. Show that $f_1$ and $f_2$ have finitely many common roots in $(\mathbb{C}^*)^2$ for any choice of coefficients. Generalize to more variables.


Solution:

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