22 problems

Let $\phi(n,d)$ denote the polynomial that gives the dimension of the space of quadrics in $n$ variables passing through $d$ general points and tangent to $\binom{n+1}{2}-d-1$ gene…

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…

Let $C_1,C_2\in\mathbb{C}^{d\times d}$ be scaling matrices, with associated delta-matroids, and let $c_1,c_2$ be the corresponding scalings. Delta-matroid invariance conjecture. If…

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

Galois-width conjecture.

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,…

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

Small-ML-degree realization conjecture. The resulting scaled Segre embedding has ML degree $d$. This gives a proposed construction realizing every degree in the indicated range; th…

Let $w$ and $w'$ be scalings for the Segre embedding of $\mathbb P^1\times\mathbb P^1\times\mathbb P^{n}$, and let $P_n$ denote the relevant polytope. For every face…

Let $v$ and $w$ be scalings for $\mathbb P^{m-1}\times\mathbb P^{n-1}$, and define … Let $M_v$ and $M_w$ be the linear matroids defined by the columns of $\widehat v$ and…

Let $A \in \mathbb{Z}^{n \times r}$ and $\overline{A} \in \mathbb{Z}^{n \times s}$ be a facial submatrix of $A$. The facial-submatrix maximum likelihood degree conjecture. The maxi…

Let $V$ be a vector space, let $\boldsymbol{S}^2V$ denote the space of symmetric matrices on $V$, and let $c6(n,a)$ be the maximum likelihood degree associated with symmetric matri…

Distinct-solutions conjecture. For generic values of $A$, $B$, and $S$, the score equations of $\mathcal{M}_{A,B}$ with sample covariance matrix $S$ have $2n-3$ distinct solutions.

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 $MA(1)$ be the moving-average model with $n>1$ sample points, and let its maximum-likelihood degree be the number of complex critical points of the likelihood function for gene…

Let $P$ be a lattice polytope. A polytope has strict linear precision when its toric blending parametrization satisfies the strict linear precision condition described in the paper…

For $i\in\{0,\ldots,m-1\}$ and $j,k\in\{0,1\}$, let $C_m$ be the no-three-way interaction model with one $m$-ary variable and two binary variables, defined by … Let $A_m$ be its as…

Let $V^c$ be the scaled Segre variety associated with an $m\times n$ scaling matrix $c\in(\mathbb C^*)^{mn}$, and let $\deg(V)=\binom{m+n-2}{m-1}$. Scaled Segre ML-degree conjectur…

Let $m\geq 2$, and let $\Lambda_m$ be the object whose coefficients are $\lambda_1,\dots,\lambda_{m-1}$. Conjectural formula. For $m\geq 2$, the first coefficient satisfies … This…

Let $X$ be an irreducible algebraic subvariety of $({\mathbb C}^*)^n$ contained in … and let $\bar X$ be its closure in $({\mathbb C}^*)^{n-1}\times{\mathbb C}$. Suppose that…

For $n\geq4$, let $\operatorname{Gr}_{2,n}\subseteq\mathbb{P}^{\binom{n}{2}-1}$ denote the Grassmannian of lines in $\mathbb{P}^{n-1}$, with homogeneous coordinates $p_{ij}$, and l…

Let $\mathcal{V}_2$ denote the variety of $3 \times n$ matrices of rank at most $2$, with $n \geq 3$. ML-degree conjecture. The ML degree of $\mathcal{V}_2$ equals … This conjectur…