82 problems

Matúš's conjecture. There exists a distribution $p$ in the zero set of $J_{\mathcal C}$ such that all joint probabilities of $p$ are rational.

Let $X$ be a closed irreducible subvariety of the hyperplane $H=\{(p_1,\ldots,p_n)\in({\mathbb{C}}^*)^n:p_1+\cdots+p_n=1\}$, and let $d=\dim X$. The Huh–Sturmfels conjecture. One h…

Generating conjecture. The ideal of homogeneous invariants for the 2-state GM+I model on $T$ is generated by those $3\times 3$ minors of edge flattenings that do not involve the va…

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 $\tau$ be a binary tree on binary random variables, and let $I_f$ be the ideal of phylogenetic invariants of its general Markov model. Each edge of $\tau$ induces a split of th…

Three-factor radical-generation conjecture. The radical of $J_{p,3}$ is the prime ideal $I_{p,3}$:

Two-factor minimal-generation conjecture. The ideal $I_{p,2}$ is minimally generated by

Linear-eliminant membership conjecture. If $m\geq 2$, $p\geq 2m+2$, and $R\cup C\neq\bar R\cup\bar C$, then

Garcia–Stillman–Sturmfels conjecture. The ideal $\mathfrak a(2;2,2,\dots,2)$ is generated by the $3\times 3$ minors of all two-dimensional flattenings arising from bipartitions of…

Bounded Markov complexity conjecture. If $\{1,2,\ldots,j\}$ is an independent subset of the underlying graph of $\Delta$, then the preceding bounded-format conclusion holds. This w…

Let $W$ and $W'$ be coefficient points for the varieties $Y_{W,n}$ and $Y_{W',n}$, respectively, and let $E_A$ denote the principal $A$-determinant, whose factors are the discrimin…

Extremal ML threshold relation. For any undirected graph $G$,

Extremal ML degree bound. For any graph $G$,

For each $n$, let $\Delta_n$ denote the relevant $n$-state model space, and let $a_n$ be the number of fundamental models in $\Delta_n$ of degree $2n-1$. For $n\geq 3$, Proposition…

Let $P$) and $Q$ be two colored paths on $m$ vertices with concentration matrices $K_P$ and $K_Q$ satisfying … A reflection of a colored path is the path obtained by reversing its…

Let $n\geq 5$. Let $\mathcal M=(G,\textit{In},\textit{Out},\textit{Leak})$ be an $n$-compartment mammillary model with $\textit{In}=\{2\}$, $\textit{Out}=\{3\}$, and…

Galois-width conjecture.

A BMT-derived graph is a graph obtained from the tree model by merging color classes of edges. Two internal nodes are non-adjacent when they are not joined by an edge in the underl…

Let $C_n$ be the cycle graph on $n$ vertices, and let $\MLdeg(C_n)$ denote the number of critical points of the Gaussian log-likelihood function for generic sample covariance data,…

Sullivant's saturation conjecture. The vanishing ideal satisfies

Adjoint non-degeneracy conjecture. For a generic $n$-gon $Q=B^T\cdot P_b$, the adjoint $\alpha_Q$ is non-degenerate, with Newton polytope equal to the Newton polytope of

Wachspress ML-degree conjecture. For a generic $n$-gon $Q$, the maximum likelihood degree is

Let $G$ be the undirected graphical model modelling a generic $n$-player game $X$ with binary choices in normal form. Let $\mathcal{C}=\operatorname{global}(G)$ and let…

Let $X_{(2,2,k)}^3$ be the no-three-way interaction model of two binary random variables and a third one with state space $k$. Its polar degrees form a sequence. Palindromicity con…

A sparse two-factor analysis model is the model associated with a graph $G$ having two latent variables, with $M_{\leq 1}(G)$ denoting the set of model polynomials indexed by subse…