On common zeros of entire functions of exponential growth

For systems of equations with an infinite set of roots, one can sometimes obtain Kushnirenko-Bernstein-Khovanskii type theorem if replace the number of roots by their asymptotic density. We consider systems of entire functions with exponential growth in the space $\mathbb C^n$, and calculate the asymptotic distribution of their common zeros in terms of the geometry of convex sets in the space $\mathbb C^n$.