32 problems

Aluffi–Harris conjecture. The same formula should admit a natural generalization to arbitrary projective varieties, including singular ones, by replacing the Euler characteristic w…

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 $1\leq i<j\leq n$, and let $\mathcal{S}_{ij}$ be the Schubert variety in ${\rm Gr}(2,n)$. ED-degree conjecture. If $i<j-1$, then … The formula was verified in the paper for…

Let $1\leq i<j\leq n$, let $\mathcal{S}_{ij}$ be the Schubert variety in ${\rm Gr}(2,n)$, and let $\Sigma_{\mathcal{S}_{ij}}$ denote its ED discriminant. Let $\mathcal{V}_{X^2}$ be…

Let $X\subseteq\mathbb{R}^n$ be an affine variety, let $K\subseteq\mathbb{R}^n$ be a generic linear subspace with $\dim K=k<n-d$, where $d=\dim X$, and let…

Let $\textit{\textbf{C}}$ be an arrangement of two generic cameras, and let $\mathcal C_{\textit{\textbf{C}},2}$ denote the corresponding conic multiview variety. Euclidean distanc…

For generic configurations of points, let $\mathcal{R}_n^{N,h}$ denote the corresponding resectioning variety, and let $\mathrm{EDD}$ denote Euclidean distance degree. Resectioning…

For generic arrangements of cameras, let $\mathcal{L}_n^{L,3,3}$, $\mathcal{L}_n^{L,3,2}$, $\mathcal{L}_n^{L^2,3,3}$, $\mathcal{L}_n^{L^2,3,2}$, $\mathcal{L}_n^{L^3,3,3}$, and…

For generic arrangements of cameras, let $\mathcal{M}_n^{N,h}$ denote the point multiview variety with $n$ cameras, ambient dimension parameters $(N,h)$, and let $\mathrm{EDD}$ den…

For a generic configuration of $n$ points, let $\mathcal{R}^{N,h}_n$ denote the corresponding point resectioning variety, and let $\mathrm{EDD}$ denote its Euclidean distance degre…

Frobenius minimal ED-degree conjecture. One has

Frobenius minimal ED-degree conjecture. In the minimization problem for the ED degree of the rank-one approximation problem over positive definite inner products $Q$, the Frobenius…

Let $X_{\bar{\mathbf{q}}, n}$ be the affine variety of image configurations arising from a camera and world points $\bar{\mathbf{q}} \in (\mathbf P^3)^n$. For generic data, let…

Let $X \subseteq \mathbb{C}^n$ be an irreducible affine variety, and let $\overline{X} \subset \mathbb{P}^n$ be its projective closure. Let $Q_{\infty}$ denote the quadric at infin…

Let $V_i$ be vector spaces with $\dim V_i=n_i+1$, and let $\mathscr{S}_{n_1,\dots,n_d}$ be the affine cone over the Segre variety of rank-one tensors in…

Let $U$ be a nonnegative tensor whose entries sum to $1$, and let $\mathcal{M}_\Sigma$ be a marginal independence model contained in the probability simplex $\Delta_{D-1}$. Let the…

Let $\mathcal{L}_2^S\subset\operatorname{Sym}_n(\mathbb{C})$ be the variety of symmetric matrices of rank at most $2$ with diagonal zero pattern $S=\{(1,1),\ldots,(s,s)\}$, where…

Let $\mathcal{L}_2^S\subset\mathbb{C}^{m\times m}$ be the variety of square matrices of rank at most $2$ with zero pattern $S=\{(1,1),\ldots,(s,s)\}$ for some $s\ge2$. Quadratic-gr…

Let $\mathcal{L}_2^S\subset\mathbb{C}^{m\times n}$ be the variety of matrices of rank at most $2$ with diagonal zero pattern $S=\{(1,1),\ldots,(s,s)\}$, where $s\in[4]$, and set…

Assume that $\mathcal{L}_r^S$ is irreducible. Let $I\subset[n]$ satisfy $|I|=m$, and let $X_{[m],I}$ and $U_{[m],I}$ denote the corresponding $m\times m$ submatrices; write…

ED-degree stabilization conjecture.

Extreme-coefficient conjecture. The extreme coefficients are

Let $X_{m,n}^d$ be a variety parameterizing a set of hypersurfaces with the same Euclidean distance degree, let $H$ be a point of $X_{m,n}^d$, and let $k$ be the number of paramete…

Let $C_m$ denote the variety associated with critical circles through a generic configuration of $m$ points in the plane, and let $EDD(C_m)$ denote its Euclidean distance degree. D…

Let $Z_m$ denote the variety of critical $(n-1)$-spheres associated with a generic configuration of $m$ points in $\mathbb R^n$, and let $EDdegree(Z_m)$ be its Euclidean distance d…