10 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…

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…

Diminishing-action conjecture. If

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

Passare–Rullgård's conjecture. A maximally sparse Laurent polynomial has a solid amoeba.