1. Critical points

Minimal and critical points
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 $r \cdot \logx$, then $p$ is critical. However this is not necessarily true when $f$ is not combinatorial.Problem 1.1.
[Steve Melczer] When do we have minimal points that are critical? 
Problem 1.2.
[Steve Melczer] If $p$ minimizes $r \cdot x$ locally on the contour, then is it critical?
Remark. [Steve Melczer] Yuliy Baryshnikov pointed out that this question is false in this level of generality. A counterexample is any contour that has cusps.


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.
Problem 1.3.
[Robin Pemantle] Is $V_f = \bigcup_{i=1}^m H_i$, where $H_i$ are smooth hypersurfaces in a small ball containing $p$? 
Problem 1.4.
[Robin Pemantle] Can computer algebra detect this? 
Symmetric multilinear functions
Let $Q(z)$ be a multivariate Laurent polynomial, $z=(z_1,\dots,z_d)$, and let $r \in \mathbb C \mathbb P^{d1}$. 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 $\mathbb C \mathbb P^{d1}$. 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 $z$).Problem 1.5.
[Robin Pemantle] Let $r=[1:\cdots:1]$, $Q$ be a symmetric multilinear function and $\delta(x)=Q(x,\dots,x)$, $\rho$ a minimum modulus root of $\delta(x)$. Then $(\rho,\dots,\rho)$ is a minimal point and a critical point in direction $r$. Can this be generalized to find critical points in other directions? 
Problem 1.6.
[Robin Pemantle] Can the multilinearity of $Q$ be removed, but keeping $r=[1:\cdots:1]$? 
Critical points at infinity
Consider a multivariate Laurent polynomial $Q(z)$, $z=(z_1,\dots,z_d)$ and $r \in \mathbb C \mathbb P^{d1}$. 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 $X \times \mathbb C \mathbb P^{d1}$.Problem 1.7.
[Robin Pemantle] If $x \in X$ is a point in the torus orbit corresponding to a face $\Gamma$ of the Newton polytope of $Q$, then is $r$ parallel to the face?
Cite this as: AimPL: Analytic combinatorics in several variables, available at http://aimpl.org/combinseveral.