Analytic combinatorics in several variables

    Suppose $f$ is a Laurent polynomial and $V_f \subseteq (\mathbb C^\times)^d$ is the subvariety of the torus. Let $M_f = (\mathbb C^\times)^d \setminus V_f$ and let $A_f$ denote the amoeba of $f$. Let $(C_\omega)_\omega$ denote the components of $\mathbb R^d \setminus A_f$. For each point $\overline \rho \in C_\omega$, let $\mathbb T_{\overline \rho}$ be the torus in $M_f$ that maps to $\overline \rho$ under $\text{Log}$, and let $[\mathbb T_{\omega}]$ denote its homology class in $H_d(M_f)$.

    Suppose we have a $d$-dimensional variety $V$, a cycle $T$ in $V$, and a basis $\sigma_i$ for $H_{2d-1}(V)$, where $\sigma_i$ are cycles “draped over the saddles". We wish to find the coefficients $n_i$ in $T=\sum_{i}n_i \sigma_i$. If $\sigma_i^{*}$ is the dual basis, then $n_i=\sigma^{*}(T)$.