83 problems

Let $A\in\mathbb{R}_{\geq 0}^{M\times N}$ have non-negative rank $H_0$, and suppose that $A=U_0V_0$ for some strictly positive matrices $U_0\in\mathbb{R}_{>0}^{M\times H_0}$ and…

Let $\Sigma$ be the alphabet, let $\mathbf{g}_{n,d}$ denote the parameterization of the relevant hidden Markov model, and let $p=(p(v))_{v\in\Sigma^n}$ be a string function. For wo…

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 $X_m$ be the affine multiview variety of $m\geq 3$ cameras in general position, and let $\operatorname{EDdeg}(X_m)$ denote its Euclidean distance degree. Multiview conjecture.…

ML-degree polynomiality conjecture. For $m\geq 3$, the ML degree is a polynomial of degree $m-1$ in $n$. In the cases stated in the source, for $m=3$ it equals

Let $C$ denote the set of small conditionals, let $J-1$ be the dimension of the underlying lattice, and consider the coefficient matrix of the Diophantine equation governing the ma…

Normality and Cohen–Macaulayness conjecture. The quotient ring

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,…