Rayleigh–Ritz Degree

• Rayleigh–Ritz degree is the count of complex critical points for a generic homogeneous polynomial constrained to the intersection of a sphere with a projective variety.
• It employs intersection theory and techniques like Chern class computations to yield explicit formulas, with applications in quantum chemistry, tensor decompositions, and numerical optimization.
• It generalizes classical eigenvalue theory by linking variational methods with the Euclidean distance degree of Veronese embeddings, providing actionable insights for algorithm design.

The Rayleigh–Ritz degree quantifies the algebraic complexity of non-linear Rayleigh quotient optimization constrained to projective algebraic varieties. For a fixed projective variety $X \subseteq \mathbb{P}^n$ and generic homogeneous polynomial $f$ of degree $\omega$, the Rayleigh–Ritz degree $RRdeg_\omega(X)$ is the number of complex critical points—projective $X$-eigenpoints—of $f$ on the intersection of the unit sphere with the affine cone over $X$. This invariant generalizes classical eigenvalue theory, connecting critical point counts in non-linear variational problems, algebraic geometry (notably the Euclidean distance degree of the Veronese embedding), and applications in quantum chemistry, tensor decompositions, and numerical optimization (Salizzoni et al., 20 Oct 2025, Borovik et al., 7 Dec 2025).

1. Foundational Concepts and Variational Setup

The classical Rayleigh–Ritz variational method addresses the minimization of the quadratic Rayleigh quotient for a self-adjoint operator $H$ on a separable Hilbert space $V$, with

$$R[\psi] = \frac{\langle \psi | H | \psi \rangle}{\langle \psi | \psi \rangle}$$

over normalized vectors $|\psi \rangle \in V$. The key principle is that $R[\psi]$ achieves its critical values at the eigenvectors of $H$, yielding upper bounds $R[\psi] \geq E_1$ (the ground-state variational principle), and more generally $R[\psi] \geq E_k$ for vectors orthogonal to the first $k-1$ eigenstates ("$k$th-level variational principle") (Fernández, 2022).

Nonlinear Rayleigh quotient optimization generalizes this scenario, replacing the quadratic form with a homogeneous polynomial $f$ of degree $\omega$, and restricting to the sphere intersected with an algebraic variety $X \subseteq \mathbb{P}^n$. The critical points of $f$ on $S^n \cap C(X)$ (where $C(X)$ is the affine cone over $X$) are termed $X$-eigenvectors, and their projective classes are $X$-eigenpoints (Salizzoni et al., 20 Oct 2025).

2. Definition and Computation of the Rayleigh–Ritz Degree

The Rayleigh–Ritz degree $RRdeg_\omega(X)$ is defined as the number of complex $X$-eigenpoints of a generic homogeneous polynomial $f$ of degree $\omega$:

$RRdeg_\omega(X) = |\{ [\psi] \in \mathbb{P}^n : [\psi]\ \text{is an $X$-eigenpoint of}\ f \}|$

for generic $f$ (Salizzoni et al., 20 Oct 2025). This count is finite and independent of the specific $f$, as established via intersection-theoretic arguments.

A central result is that $RRdeg_\omega(X)$ equals the Euclidean distance degree (ED degree) of the Veronese embedding $\nu_\omega(X)$ of $X$:

$RRdeg_\omega(X) = \mathrm{DD}(\nu_\omega(X))$

This identification establishes a deep link between critical point theory for polynomial optimization and metric algebraic geometry (Salizzoni et al., 20 Oct 2025).

To determine $RRdeg_\omega(X)$ for specific classes of varieties, intersection-theoretic techniques such as Porteous' formula, Chern class expressions, and determinantal locus counts are employed. For example, for a generic complete intersection $$X = \{f_1 = \cdots = f_c = 0\} \subseteq \mathbb{P}^n$$ of codimension $c$ and degrees $\delta_1,\dots,\delta_c$, Theorem 3.1 of (Salizzoni et al., 20 Oct 2025) provides an explicit sum formula for $RRdeg_\omega(X)$.

3. Illustrative Cases and Explicit Formulas

Closed-form expressions for the Rayleigh–Ritz degree are known for several important varieties:

$X$ (Variety)	$\omega$	$RRdeg_\omega(X)$ (Formula)
$\mathbb{P}^n$ (projective)	$2$	$n+1$
$\mathbb{P}^1$ (binary forms)	$\omega$	$\omega + 2$
Plane curve of degree $\delta$	$\omega$	$\delta(\delta+\omega-1)$
Smooth surface deg. $\delta$	$\omega$	$\delta[(\delta-1)^2 + \omega\delta + (\omega-1)^2]$
Segre—rank-1 tensors ($k$)	$\omega$	$\omega^k\ k!$
Rank-1 $n\times m$ matrices	$2$	$$\sum_{i=1}^{\min\{n,m\}} \binom{n}{i}\binom{m}{i}4^{i-1}$$

For tensor network varieties, notably Segre and tensor train (TT) varieties, these formulas are further specialized:

• Rank-one binary tensors: $RRdeg((\mathbb{P}^1)^n) = 2^n n!$
• For TT varieties that are Segre products, the RR degree can be computed combinatorially by decomposing according to the product structure (Borovik et al., 7 Dec 2025).

In numerical studies of TT varieties and low-rank determinantal loci, HomotopyContinuation.jl is used to empirically compute RR degrees, confirming these counts and providing tables for small parameters (e.g., $2 \times 2$ TT variety of rank $(1,2,1)$ has $RRdeg = 352$) (Borovik et al., 7 Dec 2025).

4. Rayleigh–Ritz Discriminant and Loci

The Rayleigh–Ritz correspondence $$\mathcal{R}_V \subseteq \mathbb{P}^{N-1} \times \mathbb{P}(\mathrm{Sym}^2\mathbb{C}^N)$$ encodes all pairs $(\psi,H)$ where $\psi$ is a critical point of $R_H|_V$, with projection generically finite of degree $RRdeg(V)$. The RR discriminant $\Sigma_V$ is the branch locus, specifying the Hamiltonians $H$ for which the number of complex critical points drops; this includes cases where critical points collide or isotropic intersections arise.

For $V = \mathbb{P}^{N-1}$, the RR discriminant coincides with the classical matrix discriminant of degree $N(N-1)$. For more complicated varieties, explicit degrees of the RR discriminant and its nonisotropic component have been tabulated numerically (e.g., for rank-one $2\times 2$ TT, the nonisotropic part has degree $6000$) (Borovik et al., 7 Dec 2025).

5. Applications in Quantum Chemistry and Tensor Network Methods

The minimization of the Rayleigh quotient over structured varieties underpins electronic structure calculations, including Hartree–Fock, CI, MCSCF, and modern tensor network ansätze. In these contexts, the Rayleigh–Ritz degree provides a measure of the algebraic complexity for finding all stationary points of the constrained eigenproblem (Salizzoni et al., 20 Oct 2025, Borovik et al., 7 Dec 2025).

The performance of iterative optimization algorithms such as ALS (alternating linear scheme) and DMRG (density-matrix renormalization group) is intimately related to the RR degree:

• ALS converges to one of the real local minima, but increasing RR degrees correspond to increasingly many complex critical points, raising the risk of trapping in high-lying local minima.
• DMRG, due to rank truncation, often solves an approximate problem and may not attain any true critical point of the fixed-rank Rayleigh quotient.
• The number of real critical points is typically much less than the (complex) RR degree.

Knowledge of $RRdeg(V_{\mathbf{k},\mathbf{r}})$ thus informs the expected landscape of local minima and provides a means for benchmarking variational strategies in computational chemistry (Borovik et al., 7 Dec 2025).

6. Broader Mathematical Significance

The Rayleigh–Ritz degree unifies diverse areas:

• It generalizes the count of classical matrix eigenvalues to eigenproblems for symmetric tensors and arbitrary projective varieties.
• Through its connection to the ED degree, it is positioned as a central object in metric algebraic geometry, making available the toolkit of intersection theory, Chern classes, and polar varieties.
• The theory engenders new explicit formulas for critical point counts in constrained polynomial optimization, with consequences for both theoretical analysis and practical computation.

A plausible implication is that advances in the calculation and interpretation of Rayleigh–Ritz degrees will further clarify the algebraic and numerical landscape of constrained optimization on varieties, especially as quantum chemistry and data science increasingly use algebraic structure in modeling and algorithm design.

7. Real vs. Complex Critical Points and Open Questions

Although the Rayleigh–Ritz degree counts complex critical points for generic data, many applications—especially in optimization and quantum computation—are limited to real stationary points. Notably, there can be significant disparities between the RR degree and the number of real solutions. For example, a plane conic $X \subset \mathbb{P}^2$ has $RRdeg_2(X) = 6$, but a real quadratic polynomial need not have six real $X$-eigenvectors (Salizzoni et al., 20 Oct 2025). This suggests that further work is required to analyze the real locus and its implications for practical algorithms.

Open research areas include the explicit computation of RR degrees for higher codimension tensor network varieties, efficient characterization of the RR discriminant, and probabilistic analysis of the real critical point count in relevant application domains.