156 problems

A Gaussian directed acyclic graph (DAG) model is the statistical model associated with a directed acyclic graph under Gaussian conditional-dependence constraints; its conditional i…

The ML-degree polynomiality conjecture. For generic $\mathcal{L}$, these ML-degrees are polynomial in $n$.

Let $X\subset\mathbb{C}\mathbb{P}^n$ be a closed irreducible subvariety with $X^\circ=X\setminus\mathcal{H}\neq\emptyset$. Its $i$-th sectional maximum likelihood degree is … where…

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.…

Let $\sigma=[n]$ with $\alpha_1=0$, and let ${\cal L}_{[n]}$ be the canonical LSSM whose parameter space consists of the symmetric matrices … Let $G_{\mathcal{L}_{[n]}}$ denote the…

Huh–Sturmfels' conjecture. The maximum likelihood degree satisfies

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…

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 $I_{p,3}$:

Circular lexicographic delightfulness conjecture. The circular lexicographic term order is $2$-delightful for $I_{p,1}$. More specifically, the reduced Gröbner basis for $I_{p,2}$…

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

Finiteness conjecture for factor-analysis ideals. For arbitrarily many factors, the ideal is generated by polynomials arising from submatrices whose size depends on the number of f…

Two-factor generation conjecture. For two-factor models, the ideal $I_{p,2}$ is generated by the $3\times 3$-minors and pentads.

Real-solution conjecture. Data with

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…

Vertex-flattening generation conjecture. For any $\kappa$ and any number of taxa $n$, the phylogenetic ideal $\mathfrak a_T$ for the general Markov model on a binary $n$-taxon tree…

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…

Cardwell's conjecture. Every GL-binomial BMT-derived model derived from a phylogenetic tree with colored nodes must be vertex regular.

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 $G$,

Let $G$ be the graph encoding conditional-independence constraints for a generic binary game with $n$ players, let $\mathcal{M}_G$ denote the discrete conditional independence mode…