197 problems

Let $K\{\mathbf{X};P\}$ be the analytic power-series ring over the valued field $K$, with $P$ the parameter set of valuation vectors, and let $I\subset K\{\mathbf{X};P\}$ be an ide…

Let $M$ and $M'$ be non-isomorphic matroids, and let $Z_M$ and $Z_{M'}$ denote their matroidal cycles in their respective ambient vector spaces. Suppose there exists an invertible…

Fix a realizable chirotope $\chi\in\{-1,1\}^{\binom{n}{3}}$. Let $\operatorname{Dr}^\chi_{3,n}$ be the chirotopal Dressian, consisting of the $\chi$-tropical Plücker vectors, and l…

Let a $(3,n)$ generalized Feynman diagram (GFD) be a combinatorial object compatible with color orderings in the sense of the paper. Compatibility-count conjecture. Every $(3,n)$ G…

Let $Y$ be a nodal curve with $b4$ nodes, let $\Gamma$ be its dual graph, and let $\mathcal B$ be the set of bonds of $\Gamma$. Define the vector-weight collection…

Let $\mathcal{T}$ be the phase space of the ultra-discrete $(g+1)$-periodic Toda lattice, let $\mathcal{C}$ be the moduli space of compact tropical curves $\Gamma_C$, and let … be…

Let $\tau$ be a binary tree on binary random variables, let $f$ be the associated multilinear map, and let $I_f$ be its prime ideal. For a $2 \times \cdots \times 2$ table, flatten…

Let $L$ be an $(n-k)$-dimensional linear subspace of $K^n$ whose tropical Plücker coordinates are all finite, and let $M\in K^{m\times n}$, with $m\geq k$, have nonzero rows in the…

Tropical halfspace characterization conjecture. A tropical polytope $P$ is pure and full dimensional if and only if it has a halfspace description whose apices are in general posit…

Tropical halfspace representation conjecture. If $P$ is pure, then the halfspaces from a generic lift $\overline{P}$ of $P$ map to tropical halfspaces whose intersection is $P$ its…

Tropical discriminant conjecture. For any point configuration $A$, the tropical discriminant $\tau(X_A^*)$ is a union of cones in the secondary fan $\Sigma(A)$.

Partition characterization. Every minimal halfspace with respect to ${\mathcal P}$ has the form

Let $L$ be a tropical linear space, and call it series-parallel when it arises from a matroidal decomposition whose facets are polytopes of series-parallel matroids. For a $d$-dime…

A tropical $d$-plane in $n$-space is a tropical linear space of dimension $d$ in $n$-space. Let $i$ be a face dimension, and map the space to the quotient by the diagonal line: … A…

Let $X\subset\mathbb{G}_{\text{m}}^d$ be an irreducible variety of dimension $r$ such that its adelic amoeba $\mathscr{A}_{\mathbb{A}}(X)$ is not contained in any hyperplane. Let…

For positive integers $m$ and $n$, let $B_{m,n}$ denote the complex of $m\times n$ matrices of Barvinok rank two. Manifold conjecture. For all $m$ and $n$, the complex $B_{m,n}$ is…

Tropical fan conjecture. If the lineality space of $\operatorname{Trop}^+ \operatorname{Spec} \mathcal{A}$ has dimension $|C|$, then the quotient of…

Cancellative-elements conjecture. An element of $S[\mathcal{M}]/\operatorname{Bend}(J)$ is cancellative if and only if it is the image of a term of $S[\mathcal{M}]$.

Normalization conjecture for tropical curves. There is an embedding of $C$ with an ideal $I$ having the property that the integral closure of $B$ in its field of fractions is obtai…

Conjectural characterization. For every $m$ with $3\leqslant m\leqslant n$, at least one of these two inequalities holds. The inequalities express a concavity-type regularity in th…

Let $d4a3^n$ be equipped with the tropical metric, and let $\mathrm{e}_{\mathrm{tr}}(\mathbb{R}^{n})$ denote the maximal cardinality of a tropical equilateral subset of $d4a3^n$. T…

Let $d4a3^n$ be equipped with the tropical metric, and let $d4a4^n$ denote its integer lattice with the induced tropical metric. The tropical chromatic number conjecture. The tropi…

Weight-Two Classification. The point

Let $p$ be a valuated matroid of rank $d$ on $\mathbb{T}^n$, let $L_p$ be its corresponding tropical linear space, and define … Here $L_p^\vee$ denotes the tropical dual of $L_p$.…

Tilde-theta evacuation conjecture. For every $T\in\operatorname{SSYT}(k,[n])$,