Rayleigh–Ritz discriminant degrees for rank-one matrices
For integers $m, n \ge 2$ let $V_{m,n} \subset \mathbb{P}^{mn-1}$ be the Segre variety of complex $m \times n$ matrices of rank one, in the entry coordinates $\psi = \operatorname{vec}(ab^{\mathsf T})$. For a complex symmetric matrix $H$ of order $mn$, consider the Rayleigh quotient $$R_H(\psi) = \frac{\psi^{\mathsf T} H \psi}{\psi^{\mathsf T} \psi}$$ on the part of $V_{m,n}$ where $\psi^{\mathsf T} \psi \ne 0$; transposes do not conjugate. The nonisotropic part of the Rayleigh–Ritz discriminant of $V_{m,n}$, $\Delta^{\mathrm{ni}}_{m,n} \subset \mathbb{P}(\operatorname{Sym}^2 \mathbb{C}^{mn})$, is the Zariski closure of the set of $[H]$, $H \ne 0$, for which $R_H$ has a degenerate critical point on $V_{m,n}$ with $\psi^{\mathsf T} \psi \ne 0$, that is, one at which its Hessian in local coordinates on $V_{m,n}$ is singular. Borovik, Friedman, Hoşten and Pfeffer computed the degrees of $\Delta^{\mathrm{ni}}_{2,n}$ for $2 \le n \le 7$ and of $\Delta^{\mathrm{ni}}_{3,n}$ for $3 \le n \le 7$ numerically, by homotopy continuation; the formulas below match these values.
Problem. For every $n \ge 2$, are $\Delta^{\mathrm{ni}}_{2,n}$ and $\Delta^{\mathrm{ni}}_{3,n}$ hypersurfaces with $$\deg \Delta^{\mathrm{ni}}_{2,n} = 24 \binom{n+1}{3} \quad \text{and} \quad \deg \Delta^{\mathrm{ni}}_{3,n} = 24 n^2 \binom{n}{2},$$ the degree being that of the reduced hypersurface?
Source: Borovik, Friedman, Hoşten and Pfeffer (2025), Conjecture 3.18 of arXiv:2512.06939v2 (Conjecture 5.18 in v1).
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- AG — Algebraic Geometry
- Secondary
- NA
- Keywords
- discriminants, OpenProblemsInNLA, Rayleigh quotients, Segre varieties
- Literature checked
- Oct. 8, 2026 by clemens-admin
References
@misc{borovik2025,
author = {Borovik, Viktoriia and Friedman, Hannah and Ho{\c{s}}ten, Serkan and Pfeffer, Max},
title = {Numerical Algebraic Geometry for Energy Computations on Tensor Train Varieties},
year = {2025},
eprint = {2512.06939},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-tr20,
author = {Townsend, Alex},
title = {{TR-20}: Rayleigh–Ritz discriminant degrees for rank-one matrices},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/tensor-computations/TR-20/README.md},
note = {Read at commit eb7c9d6, 2026-10-07},
}
Resolution claims
A logarithmic compactification of the critical-point correspondence on $\mathbb{P}^{m-1} \times \mathbb{P}^{n-1}$ gives a coefficient formula for its nonisotropic Rayleigh–Ritz discriminant. Its derivation counts ramification on a general parameter pencil, computes the boundary contributions by first-jet bundles, and gives an argument that the nonisotropic ramification maps generically one-to-one onto a reduced irreducible hypersurface. The formula yields $24\binom{n+1}{3}$ for $m = 2$ …
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-00213, "Rayleigh–Ritz discriminant degrees for rank-one matrices", clopen.solutions.