The Euclidean distance degree of the real symplectic group
For $m \ge 1$ let $$J = \begin{pmatrix} 0 & I_m \\ -I_m & 0 \end{pmatrix}, \qquad \mathrm{Sp}_{2m}(\mathbb{C}) = \{X \in \mathbb{C}^{2m \times 2m} : X^{\mathsf T} J X = J\},$$ and let $D_m$ be the Euclidean distance degree of the real symplectic group for the Frobenius inner product: the number of complex critical points on $\mathrm{Sp}_{2m}(\mathbb{C})$ of $$f_U(X) = \operatorname{tr}\bigl((X - U)^{\mathsf T}(X - U)\bigr)$$ for generic $U \in \mathbb{C}^{2m \times 2m}$, that is, for $U$ outside a proper Zariski-closed set. These critical points are the $X$ with $X^{\mathsf T} J X = J$ and $\operatorname{tr}\bigl((U - X)^{\mathsf T} X H\bigr) = 0$ for every $H$ with $H^{\mathsf T} J + J H = 0$. The transpose is not a conjugate transpose, so $f_U$ is the complexification of the squared Frobenius distance. Baaijens and Draisma note that the count may depend on how the symplectic form sits relative to the inner product; the problem uses the standard pair above. They computed $D_1 = 4$, $D_2 = 24$ and $D_3 = 544$, which agree with the formula below.
Problem. Is $$D_m = 2^{m^2} + 2^{2m-1}$$ for every $m \ge 1$?
Source: Baaijens and Draisma (2015), Problem 5.1, stated there as determining the degree; the formula is the pattern they suggest just before it. In this form, OpenProblemsInNLA.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- NA — Numerical Analysis
- Secondary
- AG, OC
- Keywords
- Euclidean distance degree, matrix nearness problems, OpenProblemsInNLA, symplectic groups
- Literature checked
- Oct. 7, 2026 by clemens-admin
References
@article{baaijens2015,
author = {Baaijens, Jasmijn A. and Draisma, Jan},
title = {Euclidean distance degrees of real algebraic groups},
journal = {Linear Algebra Appl.},
volume = {467},
year = {2015},
pages = {174--187},
doi = {10.1016/j.laa.2014.11.012},
eprint = {1405.0422},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-sp03,
author = {Townsend, Alex},
title = {{SP-03}: The Euclidean distance degree of the real symplectic group},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/eigenvalues-and-inverse-problems/SP-03/README.md},
note = {Read at commit 7336302, 2026-10-06},
}
Resolution claims
This report studies the Euclidean distance degree $D_m$ of the standard real symplectic group, for which Baaijens and Draisma suggested the formula $D_m=2^{m^2}+2^{2m-1}$. Building on an earlier regular $m^2$-variable reduction, we identify its denominator exactly and put the full generic critical-point problem on a universal discrete-Lyapunov domain, where the two auxiliary symmetric variables solve explicit Stein equations. We compute the …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-09 | Open → Partially resolved | The registry's curators | Stein normal form and critical points v1 recorded from OpenProblemsInNLA @ eb7c9d6, SP-03/report.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00154, "The Euclidean distance degree of the real symplectic group", clopen.solutions.