Analytic combinatorics in several variables

    Suppose $f(x)$ is a (multivariate) Laurent polynomial, and we are looking at its Laurent series expansion corresponding to a component $C$ of the complement of the amoeba $A_f$. If $f =\sum_{\omega} a_\omega x^{\omega}, a_\omega \in \mathbb Q_{\geq 0}$ (that is $f$ is combinatorial), and $p$ is a minimal point that is a minimizer of the height function $-r \cdot \log|x|$, then $p$ is critical. However this is not necessarily true when $f$ is not combinatorial.

    Let $f(x_1,\dots,x_d)$ be a polynomial. Suppose $p$ is a point such that $f(p)=\frac{\partial f}{\partial x_j}(p)=0$ for $j=1,\dots,d$. Let $V_f$ denote the variety.

    Let $Q(z)$ be a multivariate Laurent polynomial, $z=(z_1,\dots,z_d)$, and let $r \in \mathbb C \mathbb P^{d-1}$. In the smooth case, $z$ is a critical point in direction $r$ if $\left(z_1 \frac{\partial Q}{\partial z_1},\dots, z_1 \frac{\partial Q}{\partial z_d}\right)=r$ in $\mathbb C \mathbb P^{d-1}$. In general, if you look at the maximal smooth piece of the variety $V_Q$ that contains $z$, then this binary relation is ($r$ is in the log normal space to $V_Q$ at $z$).

    Consider a multivariate Laurent polynomial $Q(z)$, $z=(z_1,\dots,z_d)$ and $r \in \mathbb C \mathbb P^{d-1}$. Let $x^{(1)},x^{(2)},\dots$ be a sequence of points in the variety $V_Q$, where $x^{(j)}$ is a critical point in the direction $r^{(j)}$. Let $X$ denote the toric variety of the Newton polytope of $Q$. Let $(x,r)$ be any subsequential limit of $(x^{(j)},r^{(j)})$ in $X \times \mathbb C \mathbb P^{d-1}$.