OP-00245 Solution claimed

Critical-point count for corank-one approximation with a fixed zero

For $n \ge 3$ let $$V_n = \{X \in \mathbb{C}^{n \times n} : \det X = 0,\ x_{11} = 0\}.$$ Its Euclidean distance degree $\operatorname{EDdeg}(V_n)$ is the number of complex critical points, on the smooth locus of $V_n$, of $$d_U(X) = \sum_{i,j=1}^{n} (x_{ij} - u_{ij})^2$$ for a generic $U \in \mathbb{C}^{n \times n}$, that is, for $U$ outside a proper algebraic subset. The distance is bilinear, without complex conjugation; for real matrices it is the squared Frobenius distance. Kubjas, Sodomaco and Tsigaridas computed $\operatorname{EDdeg}(V_n)$ numerically for $3 \le n \le 10$, obtaining $8, 13, 18, 23, 28, 33, 38, 43$. For $n = 2$ the variety is the union of two linear spaces, and its Euclidean distance degree is $2$.

Problem. Is $\operatorname{EDdeg}(V_n) = 5n - 7$ for every $n \ge 3$?

Source: Kubjas, Sodomaco and Tsigaridas (2022), Conjecture 5.1, stated there as 5(n − 1) − 2 without a range of n; OpenProblemsInNLA requires n ≥ 3.

Imported from github.com/ajt60gaibb/OpenProblemsInNLA.

Classification

Primary: AG — Algebraic Geometry
Secondary: NA, OC
Keywords: determinantal varieties, Euclidean distance degrees, low-rank approximations, OpenProblemsInNLA
Literature checked: Oct. 8, 2026 by clemens-admin

References

@article{kubjas2022,
  author        = {Kubjas, Kaie and Sodomaco, Luca and Tsigaridas, Elias},
  title         = {Exact solutions in low-rank approximation with zeros},
  journal       = {Linear Algebra Appl.},
  volume        = {641},
  year          = {2022},
  pages         = {67--97},
  doi           = {10.1016/j.laa.2022.01.021},
  eprint        = {2010.15636},
  archivePrefix = {arXiv},
}
@misc{openproblemsinnla-ra19,
  author       = {Townsend, Alex},
  title        = {{RA-19}: Critical-point count for corank-one approximation with a fixed zero},
  howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
  year         = {2026},
  url          = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/randomized-and-low-rank-approximation/RA-19/README.md},
  note         = {Read at commit eb7c9d6, 2026-10-07},
}

Resolution claims

Critical-point count for corank-one approximation with one fixed zero v1 GitHub ↗

George Stepaniants

The manuscript proves that for every $n \ge 3$ the variety of singular complex $n \times n$ matrices with $x_{11} = 0$ has Euclidean distance degree $5n - 7$ for the bilinear squared Frobenius distance, counting critical points on its smooth locus for generic complex data. After an orthogonal reduction of the data, the characteristic polynomial of $A(z)A(z)^{\mathsf T}$, where …

Full resolution · not verified · fidelity not established · posted on GitHub 12 Sep 2026

Status history

When	Change	By	Reason
2026-10-08	Needs verification → Solution claimed	clemens-admin	Listed with the record of its solution, unreviewed.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00245, "Critical-point count for corank-one approximation with a fixed zero", clopen.solutions.