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1. Critical points

    1. 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 $-r \cdot \log|x|$, 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?
            1. Remark. [Steve Melczer] Yuliy Baryshnikov pointed out that this question is false in this level of generality. A counter-example 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?
                          If this is true, then it is much easier to compute asymptotics. Some of the generating functions that appear in random tilings are of this kind.
                        • Symmetric multilinear functions

                              Let $Q(z)$ be a multivariate Laurent polynomial, $z=(z_1,\dots,z_d)$, and let $r \in \mathbb C \mathbb P^{d-1}$. 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 $z$).

                          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 $r \in \mathbb C \mathbb P^{d-1}$. 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 $X \times \mathbb C \mathbb P^{d-1}$.

                                  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.