Rayleigh-Ritz Discriminant in Algebraic Varieties

• Rayleigh-Ritz Discriminant is a projective subvariety that identifies symmetric matrices causing a drop in the expected number of isolated critical points.
• It is computed by analyzing the rank drop in the Jacobian of the critical ideal, linking algebraic geometry with constrained energy minimization.
• Its decomposition into isotropic and non-isotropic parts provides insights into computational methods used in quantum chemistry and tensor analysis.

The Rayleigh-Ritz discriminant arises in the context of critical point analysis for Rayleigh quotient minimization problems over projective algebraic varieties. This discriminant identifies the locus of symmetric matrices (or, in broader contexts, homogeneous polynomials) for which the expected number of isolated complex critical points—termed the Rayleigh-Ritz degree—drops below the generic value. Its study is central to the enumerative, geometric, and computational aspects of constrained energy minimization and eigenvalue-type problems on algebraic varieties, with applications ranging from quantum chemistry to tensor analysis (Borovik et al., 7 Dec 2025, Salizzoni et al., 20 Oct 2025).

1. Rayleigh-Ritz Degree and Critical Point Formulation

Given an irreducible complex projective variety $V \subset \PP^{n-1}$ of codimension $c$, the Rayleigh quotient associated with a symmetric matrix $H \in \Sym^2(\C^n)$ is given by

$R_H(\psi) = \frac{\psi^T H\,\psi}{\psi^T\psi}.$

The stationary points of $R_H$ restricted to $V$ correspond algebraically to solutions of

$$\nabla_\psi R_H(\psi) \perp T_\psi V,\quad \psi^T\psi \neq 0,$$

where $T_\psi V$ denotes the tangent space to $V$ at $\psi$. For generic $H$, this system has finitely many complex solutions in the regular locus $V_{\text{reg}}$, and their number, denoted $\deg_{\rm RR}(V)$, is the Rayleigh-Ritz degree of $V$ (Borovik et al., 7 Dec 2025). The critical ideal formulation enables algebraic computation of these points: $I_{\rm crit}(V,H) = \big( I_V + \langle (c+1)\text{-minors of} \; [(\psi^T\psi)\psi^T H - (\psi^T H\psi)\psi^T\ |\ \Jac(I_V)] \rangle \big) : (I_{V_{\rm sing}} + \langle \psi^T\psi \rangle)^\infty,$ where $I_V$ is the defining ideal of $V$.

2. Rayleigh-Ritz Discriminant: Definition and Structure

The Rayleigh-Ritz discriminant, denoted $\Sigma_V$, is the projective subvariety of $\PP(\Sym^2 \C^n)$ consisting of all symmetric matrices $H$ for which the number of isolated complex critical points of $R_H|_V$ is strictly less than $\deg_{\rm RR}(V)$: $\Sigma_V = \mathrm{closure}\left\{ H \in \PP(\Sym^2 \C^n) \mid \#\{\text{critical points of } R_H|_V\} < \deg_{\rm RR}(V) \right\}.$ Algebro-geometrically,

$\Sigma_V = \pi_2 \left\{ (\psi, H) \in \RR_V : \operatorname{rank}_\psi \Jac(\RR_V) \leq n-2 \right\},$

where $\RR_V$ is the incidence variety of critical pairs and $\pi_2$ is projection onto the parameter $H$ (Borovik et al., 7 Dec 2025). For generic $V$, $\Sigma_V$ is a hypersurface whose defining equation vanishes precisely where the fiber cardinality of critical points drops.

3. Decomposition: Isotropic and Non-Isotropic Parts

The discriminant $\Sigma_V$ admits a decomposition into "isotropic" and "non-isotropic" components:

• Isotropic Part: Corresponds to matrices admitting isotropic critical points (i.e., critical points where $\psi^T\psi = 0$). Empirically, all irreducible isotropic factors are sums of squares and do not contribute to real degeneracy in the energy minimization count [(Borovik et al., 7 Dec 2025), Proposition 4.26, Conjecture 4.27].
• Non-Isotropic Part: Governs true degeneracies of the critical locus relevant for the deficiency in complex critical points. Its vanishing locus dominates the count of critical points in both the complex and real settings.

4. Algebraic-Geometric Correspondence and Computational Framework

The Rayleigh-Ritz discriminant is constructed via the RR correspondence: $\RR_V = \overline{\{ (\psi, H) \in V_{\rm reg} \times \PP(\Sym^2\C^n) \mid \psi \text{ is critical for } R_H \}} \subset \PP^{n-1} \times \PP^{N-1},$ where $N = \binom{n+1}{2}$. The projection $\pi_2: \RR_V \to \PP^{N-1}$ is generically finite with the generic fiber cardinality $\deg_{\rm RR}(V)$. The discriminant hypersurface $\Sigma_V$ is cut out by the locus where the differential of this projection drops rank, that is, where critical points fail to be isolated.

For varieties with birational parametrizations—such as Segre products or special tensor train varieties—explicit rational parameterizations enable effective formulation of the critical-point equations. For arbitrary tensor train ranks and higher-dimensional determinantal varieties, homotopy continuation and numerical algebraic geometry are required to compute $\deg_{\rm RR}(V)$ and sample $\Sigma_V$ (Borovik et al., 7 Dec 2025).

5. Explicit Examples and Degree Formulas

Closed-form degree expressions for $\deg_{\rm RR}(V)$ exist in several classical cases:

Variety Type	Example	Degree Formula/Value
Hypersurface	Degree $d$ in $\PP^{n-1}$	$d\sum_{j=1}^{n-1}j(d-1)^{n-1-j}$
Segre Product	$(\PP^1)^n$ (binary rank-one)	$2^n n!$
Segre Product	$\PP^{n-1} \times \PP^{m-1}$	$\sum_{i=1}^n 4^{i-1}\binom{n}{i}\binom{m}{i}$
Determinantal Variety	Rank-1, $2\times 2$	$\deg_{\rm RR} = 4$
Tensor Train (TT)	Order-3, TT-ranks (1,2,1)	$\deg_{\rm RR} = 32$

In computed cases, the degree of the nonisotropic part of $\Sigma_V$ was found, for example, to be 24 for $2 \times 2$ rank-one matrices, and 96 for $2 \times 3$ rank-one [(Borovik et al., 7 Dec 2025), Table 5.3].

6. The Rayleigh-Ritz Discriminant in Broader Optimization Settings

In generalizations involving the minimization of arbitrary homogeneous polynomials $f$ (beyond the quadratic case), the Rayleigh-Ritz degree quantifies the number of $X$-eigenpoints of $f$ for a projective variety $X \subset \PP^n$ (Salizzoni et al., 20 Oct 2025). The constancy of this count for generic $f$ is equivalent to the generic non-vanishing of the associated discriminant. The discriminant thus generalizes as the locus of polynomials where the associated KKT system no longer defines a zero-dimensional complete intersection.

This invariant is intimately related to the Euclidean distance degree of the Veronese embedding of $X$; the discriminant thus governs both geometric optimality conditions and the stability of enumerative critical point counts under perturbations of the energy functional.

7. Conclusions, Conjectures, and Open Directions

The Rayleigh-Ritz discriminant delineates parameter regimes of degeneracy in energy minimization on algebraic varieties. Its explicit description remains tractable only in special cases; for general tensor settings or complex parameterizations, only numerical or partial structural knowledge is currently available. Important conjectures include:

• For the rational normal curve $C_d \subset \PP^d$, the nonisotropic discriminant degree is conjectured to be $6(d-1)$, with empirical verification up to $d \leq 10$.
• Every irreducible isotropic factor is conjectured to be a sum of squares, reflecting the absence of real degeneracies in generic cases (Borovik et al., 7 Dec 2025).

The computational and theoretical study of the Rayleigh-Ritz discriminant continues to underlie benchmarks in quantum chemistry, tensor-based optimization, and related fields where the algebraic structure of constrained eigenvalue-type problems is paramount (Borovik et al., 7 Dec 2025, Salizzoni et al., 20 Oct 2025).