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.


Partial Solution:

Suppose that $f_1$ and $f_2$ have infinitely many common roots in $(\mathbb{C}^*)^2$, then we must have that $f_1$ and $f_2$ share a common factor $h$ whose variety span a subspace of dimension greater than or equal to $1$. In particular, since there are infinitely many common roots in $(\mathbb{C}^*)^2$, we have that $h$ is not a monomial, so $\dim(\textrm{Newt}(h))>0$. Writing $f_1 = h\cdot g_1$ and $f_2 = h\cdot g_2$ for some $g_1,g_2 \in \mathbb{C}[x,y]$, we have that $\textrm{Newt}(f_1) = \textrm{Newt}(h) + \textrm{Newt}(g_1)$ and $\textrm{Newt}(f_2) = \textrm{Newt}(h) + \textrm{Newt}(g_2)$. In the Minkowski sum $\textrm{Newt}(h) + \textrm{Newt}(g_1)$, we have edges parallel to the edges of $\textrm{Newt}(h)$ and similarly for $\textrm{Newt}(h) + \textrm{Newt}(g_2)$, thus $\textrm{Newt}(f_1)$ and $\textrm{Newt}(f_2)$ must have edges with common slopes.

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