Symmetric Persistent Tensors

• Symmetric persistent tensors are homogeneous polynomials defined by a recursive differentiation property that ensures non-degeneracy at every step.
• Their structure is characterized by a unique Hessian determinant factorization, providing an effective algebraic criterion tied to invariant theory and projective geometry.
• Explicit normal forms in low dimensions underpin applications in quantum information theory and algebraic complexity by reinforcing robust entanglement properties.

Symmetric persistent tensors are a recursively defined class of homogeneous polynomials (equivalently, symmetric tensors) in finite-dimensional complex vector spaces, characterized by a structural robustness under successive differentiation. These objects are central to the algebraic paper of stability in tensor rank, connections to quantum information theory, and classical projective geometry. Their defining property is intimately linked to the highly degenerate factorization of the Hessian determinant of the associated polynomial, yielding a concrete and effective algebraic criterion for persistence (Gharahi et al., 8 Oct 2025).

1. Recursive Definition and Fundamental Properties

A symmetric tensor $f \in \mathrm{Sym}^n \mathbb{C}^d$ is called persistent if it satisfies a recursive differentiation-based criterion:

• For $n=2$ (quadratic case), $f$ is persistent if it is nonsingular (i.e., its associated matrix is invertible, and no redundant variable appears).
• For $n > 2$, $f$ is persistent if it is concise (not a cone and involves all directions in $V$) and there exists a hyperplane $S \subsetneq V^\vee$ such that for every $u \notin S$, the directional derivative $\partial f / \partial u$ is persistent as a symmetric tensor of degree $n-1$.

Explicitly, conciseness requires that the polynomial cannot be written in fewer than $d$ variables: it is not linearly degenerate. The persistence property ensures that after any generic differentiation, the resulting tensor (now of degree $n-1$) remains, recursively, persistent, up to a limiting quadratic case. Thus, symmetric persistent tensors are those that "stick together" under sequential differentiation, never degenerating into a singular or reducible form at any order.

2. Hessian Determinant Characterization

A central algebraic characterization emerges via the Hessian determinant $\det H_f$ of $f$. For $f \in \mathrm{Sym}^n \mathbb{C}^d$ of degree $n$, the Hessian matrix $H_f$ is the $d \times d$ matrix of second partial derivatives: $$(H_f)_{ij} = \frac{\partial^2 f}{\partial x_i \partial x_j}$$.

The core criterion is:

• $f$ is persistent $\implies$ there exists a nonzero linear form $\ell$ such that

$\det H_f = \ell^{d(n-2)}$

• For $n = 3$ (cubic tensors) or $d \leq 3$, this condition is also sufficient: the converse holds, establishing equivalence.

This means the entire Hessian determinant of $f$, a homogeneous polynomial of degree $d(n-2)$, factors as the $d(n-2)$-th power of a nonzero linear form, reflecting an extreme degeneracy in second-order behavior. The factorization expresses the anomaly that for a persistent tensor, all curvature information is concentrated along a single direction (the kernel of $\ell$).

Relatedly, the paper proves that the Hessian of a symmetric persistent tensor always further factors as the $d$-th power of a form of degree $(n-2)$, i.e.,

$\det H_f = g^d$

for some nonzero $g$ homogeneous of degree $n-2$. For cubics or dimensions $d \leq 3$, these criteria coincide.

3. Explicit Normal Forms and Classification

The persistent property allows for a classification of normal forms in small dimensions:

• For $d = 2$, any persistent symmetric tensor is (up to a $GL(2,\mathbb{C})$ transformation) equivalent to the monomial $x_0^{n-1} x_1$.
• For $d = 3$, a generic persistent form is linearly equivalent to

$f = x_0^{n-1} x_2 + x_0^{n-2} x_1^2$

which geometrically corresponds to a line with high multiplicity plus a tangent conic.

These normal forms encapsulate the persistent structure: their behavior under differentiation produces, recursively, new forms of the same structural type.

4. Algebraic and Geometric Context: Prehomogeneous Geometry, Semi-invariants, Homaloidal Polynomials

Persistent polynomials with this Hessian property often serve as basic semi-invariants in prehomogeneous vector spaces under parabolic group actions. The paper shows that the hypersurfaces defined by persistent polynomials typically arise as relative invariants for non-reductive group actions, such as certain parabolics or solvable subgroups. The Perazzo hypersurface in $\mathbb{C}\mathbb{P}^4$ is analyzed in this context.

All persistent cubics are shown to be homaloidal: their polar map (Gauss map, gradient flow)

$$x \mapsto \left(\frac{\partial f}{\partial x_0}: \ldots : \frac{\partial f}{\partial x_{d-1}}\right)$$

is birational. This is directly linked to the unique factorization of the Hessian and the non-degeneracy of the persistent property under gradient flow—the persistence ensures the fiber of the polar map above a generic point is a singleton.

Explicit computations of Legendre transforms show that the polar/gradient inversion associated to persistent polynomials is rational and regular, further indicating the geometric rigidity of these objects.

5. Implications in Quantum Information Theory and Algebraic Complexity

Persistent tensors (in the sense established here) are inspired by quantum information theory; they generalize the "W state" tensors for qubits and $d$-level systems (qudits), yielding multipartite entangled states that are robust to loss of particles. Specifically, these tensors provide nontrivial lower bounds on tensor rank: if $f$ is persistent, then when one contracts over any subset of indices (i.e., "tracing out" qudits), the resulting tensor remains nontrivial, i.e., does not degenerate to a less entangled or decomposable form.

This recursive robustness translates to concrete lower bounds in algebraic complexity for the associated polynomials. Moreover, the persistence/hessian-determinant characterization ties to the classical Gordan–Noether–Segre theory of hypersurfaces with degenerate Hessians, implying connections to dual varieties and classical projective geometry.

6. Relation to Other Notions and Broader Impact

The effective criterion via the Hessian determinant provides an "algorithmic" test of persistence. The persistence property is strictly more restrictive than general symmetry: very few symmetric tensors have such highly degenerate Hessians. However, in the cubic (or low-dimensional) case, the algebraic criterion is both necessary and sufficient and leads directly to a finite list of normal forms. This provides both a practical recognition procedure and a structural understanding of the landscape of persistent polynomials.

By connecting persistence to homaloidal polynomials and their Legendre transforms, the paper brings together algebraic geometry, invariant theory, and applications in quantum entanglement. The structural insights into the robust entanglement properties of symmetric tensors have potential implications for quantum computation and many-body systems, while the links to classic algebraic geometry and semi-invariants invite further research in invariant theory and moduli of projective hypersurfaces.