24 problems

Let $V_d=\{f_{1,d}=\cdots=f_{r,d}=0\}\subset(\mathbb C^\ast)^n$ be the complete intersection under consideration, and let $\Delta_{i,d}$ be the Newton polytope of $f_{i,d}$. Set…

Passare–Rullgård's solid amoeba conjecture. The amoeba of a maximally sparse polynomial should be solid, meaning that

Let $V$ be a subvariety of the torus $(\mathbb{C}^\times)^n$ of codimension $k{+}1$, and let its amoeba be the image under the coordinatewise map $z\mapsto \log|z|$. A subset…

Let $Y\subseteq ({\mathbb{C}}^\ast)^{n-1}$ be a smooth algebraic hypersurface, and define … Let…

Let $V_d=\{f_{1,d}=\cdots=f_{r,d}=0\}\subset(\mathbb C^\ast)^n$ be a sequence of smooth nondegenerate complete intersections with $\deg(f_{i,d})=d_i(d)\to\infty$. Assume that the l…

Let $G/H$ be a spherical homogeneous space, let $Y\subseteq G/H$ be a subvariety, and let $\operatorname{sLog}_{\Gamma,t}(Y)$ be its spherical amoeba and $\operatorname{strop}(Y)$…

Let $M$ be a loopless matroid on a finite ground set $E$, and let $4\Sigma(M) \subseteq \mathbb{R}^E$ be its matroid fan. For a rational subspace $R \subseteq \mathbb{R}^E$, define…

Let $P(z,w)=k-p(z,w)$ be a Newton polynomial with fixed $p(z,w)$. Let $m_l(P)$ and $m_g(P)$ denote the liquid and gas contributions to the Mahler measure, and let $\mathcal{A}(P)$…

Let $P(\boldsymbol{z})=k-p(\boldsymbol{z})$ be a Laurent polynomial in $\boldsymbol{z}\in\mathbb{C}^n$, and let $\mathcal{A}(P)$ be its amoeba. Write $V_\textup{bdd}$ for the volum…

Let $P(z,w)=k-p(z,w)$ be a Newton polynomial, and let $R_k(x,y)$ denote its Ronkin function at flow parameter $k>0$. For $k_2>k_1$, distinguish the non-linear, bounded linear, and…

Let $P(z,w)=k-p(z,w)$ be a Newton polynomial undergoing the Mahler flow, with Mahler measure $m(P)$ and flow parameter $k>0$. Mahler-flow monotonicity conjecture. The Mahler measur…

A dimer model has a slope $(n_x,n_y)$ and entropy density $\sigma(n_x,n_y)$ defined on its Newton polygon $NP(P)$; phase transitions occur at boundaries of the associated Amoeba, w…

Diminishing-action conjecture. If

Let $V$ be a subvariety of the algebraic torus $({\mathbb C}^\times)^n$. Its amoeba is the image of $V$ under the logarithmic moment map, and $V$ has a finite amoeba basis if its a…

Let an amoeba or tropical variety of dimension $k$ be contained in $\mathbb{R}^n$, and consider its complement. Nisse–Sottile's conjecture. The complement of every $k$-dimensional…

Let $f(x_1,\ldots,x_n)\in\mathbb{C}[x_{1}^{\pm 1},\ldots,x_{1}^{\pm 1}]$ be a Laurent polynomial. Let $\mathcal{N}_f$ be its Newton polytope, let…

Homological independence conjecture. The toric cycles $\Gamma_\nu$ constitute a homologically independent family in the homology group $H_n((\mathbb{C}^{\times})^n\setminus V)$.

Let $A$ be the finite support of a Laurent polynomial $f$ and let $\boldsymbol{T}^n$ be the real $n$-torus. The coamoeba $\mathcal{C}_f$ is the image of the zero set of $f$ in…

A circuit polynomial is a polynomial whose support is a circuit. From the viewpoint of amoeba theory, Theorem AmoebaSolidness concerns the solidness of amoebas associated with such…

Let $f$ be an $n$-variate $t$-nomial, and let $\mathrm{Amoeba}(f)$ and $\mathrm{ArchTrop}(f)$ denote its amoeba and Archimedean tropicalization. For subsets…

Amoeba conjecture. $U_n$ is the $2^n$-th part of the amoeba of a certain Laurent polynomial, and the Newton polytope of this polynomial is the union, with the central point…

Let $A$ be a maximally sparse configuration, and consider the closed coamoeba associated with $A$ and coefficients of its defining polynomial. Passare's conjecture. The maximal num…

Gross's geometric conjecture. The fibration is piecewise smooth; $\Sigma$ is a complex one-dimensional subvariety whose singularities are transverse triple points locally given by…

Let $N_{\rm C}$ and $N_{\rm F}$ satisfy $N_{\rm F}>N_{\rm C}$, and let $H_0$ be the moduli matrix of the theory. In the strong gauge coupling limit, define the components … where…