Applications

1. From number theory: Consider one polynomial in one variable with integer coefficients, written as a straight line program of complexity $\tau$. How many roots are there which are congruent to 1 modulo $p$, as a function of $\tau$ ?

   There exist estimates using amoeba theory, by work of Maurice Rojas.

2. Prove Calabi conjecture for tropical Calabi-Yaus: in each metric class there is a unique metric which solves the tropical Monge-Ampère measure.

   In the real case this corresponds to $\det(\text{Hessian}) = \text{const}$.

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