Persistent Polynomials: Structure & Significance

• Persistent polynomials are homogeneous forms defined by recursive nondegeneracy and differential constraints, ensuring their structure persists under successive differentiation.
• They are characterized by a unique Hessian factorization into a pure power of a linear form, linking them to invariant theory and minimal tensor entanglement.
• Their classification in low dimensions and algorithmic detectability offer practical insights for applications in algebraic geometry, tensor rank analysis, and quantum information.

A persistent polynomial is a homogeneous polynomial whose algebraic and geometric structure is uniquely constrained by recursive differential stability conditions—originally motivated by the paper of persistent tensors and minimal entanglement structures in quantum information theory—and is characterized by an explicit factorization property of its Hessian determinant. Persistent polynomials occupy a special locus in the space of symmetric tensors, serving as a bridge between algebraic geometry, invariant theory, and multilinear complexity.

1. Recursive Definition and Core Characterization

Let $f \in \mathrm{Sym}^n \mathbb{C}^d$ be a degree-$n$ homogeneous polynomial in $d$ variables. The persistence property is imposed via a recursive criterion:

• For $n=2$ (quadratic forms), $f$ is persistent if it is concise, i.e., its associated symmetric bilinear form is nondegenerate (no variable is redundant).
• For $n>2$, $f$ is persistent if it is concise and, moreover, there is a hyperplane $S \subset (\mathbb{C}^d)^\vee$ such that for every linear form $L \not\in S$, the directional derivative $\partial_L f$ (a degree $n-1$ form) is persistent.

This recursive scheme ensures that the nondegeneracy and richness of algebraic structure "persist" under successive directional differentiations, generalizing the classical notion of a nondegenerate quadratic.

2. Hessian Condition and Algebraic Criterion

The central result is an explicit algebraic criterion for persistence in terms of the Hessian determinant

$$\operatorname{Hess}(f) = \det \left( \frac{\partial^2 f}{\partial x_i \partial x_j} \right).$$

For $f \in \mathrm{Sym}^n \mathbb{C}^d$, the Hessian is a homogeneous polynomial of degree $d(n-2)$. The pivotal theorem states:

• $f$ is persistent if the Hessian factors as

$\operatorname{Hess}(f) = \ell^{d(n-2)}$

for some nonzero linear form $\ell$.

Complete characterization via the Hessian holds (i) for cubic forms ($n=3$) in arbitrary dimension and (ii) for arbitrary degree in $d \leq 3$, using classical results of B. Segre and associated invariant theory. The converse—that this factorization property suffices for persistence—also holds in these cases.

For higher degree and/or dimension, further equivalences are established:

• Persistence $\Longleftrightarrow$ a certain multihomogeneous polarization ("partial polarization") of $f$ is a $d$-th power (i.e., $f$ polarizes under $(n-2)$ differentiations to a form whose Hessian is again a $d$-th power of a linear form).
• This, in turn, implies the existence of some homogeneous $g$ such that $\operatorname{Hess}(f) = g^d$.

3. Normal Forms and Classification in Small Dimensions

The explicit classification of persistent polynomials is established in "small" cases:

• $n=3$, $d=2$: Unique up to $\mathrm{GL}(2)$ transformations; normal form is $x_0^2 x_1$.
• $n=3$, $d=3$: Normal form (up to $\mathrm{GL}(3)$) is $x_0^1(x_0 x_2 + x_1^2)$ (interpreted as a line and tangent conic counted with multiplicity).
• $n=3$, $d=4$: Normal forms include cubic expressions of the type $$f = \lambda_1 x_0^2 x_3 + \lambda_2 x_0 x_1 x_2 + \lambda_3 x_1^3$$, with $\lambda_1 \lambda_2 \neq 0$ and corresponding to a Hessian that is the fourth power of a linear form.

The paper uses polarization and classical invariant theory to identify and reduce persistent polynomials to such canonical representatives, describing their orbit structure under the general linear group and their families within the algebraic landscape.

4. Connections to Prehomogeneous Geometry, Homaloidal Polynomials, and Invariant Theory

Persistent polynomials admit deep relationships with other central objects in algebraic geometry and invariant theory:

• Prehomogeneous Vector Spaces: Persistent polynomials populate loci equipped with nontrivial semi-invariants—polynomials transforming by a character under the action of $\mathrm{GL}(d)$—and thus lie in the intersection of representation theory and geometry.
• Homaloidal Polynomials: All cubic persistent polynomials are shown to be homaloidal. Homaloidal polynomials are defined by the property that their gradient (or polar map) is birational, a strong algebro-geometric constraint tied to the theory of Cremona transformations.
• Legendre Transform: The polar map and its inverse (the Legendre transform or multiplicative Legendre transform) play a role in exhibiting the relationships between persistent polynomials, homaloidal hypersurfaces, and the explicit inversion of the projective gradient map.

5. Structural Properties and Algorithms

An effective algorithmic detection of persistence follows from the main Hessian criterion. For given $f \in \mathrm{Sym}^n \mathbb{C}^d$, one computes the Hessian determinant and verifies if it factors as a pure power of a nonzero linear form. In the affirmative case, by the chain of implications, $f$ is persistent. The method of multihomogeneous polarization reduces the analysis of persistence to checking power-factoring of lower-degree forms, rendering the criterion computationally tractable for small $n$ and $d$.

In all cases, the persistence property implies high isotropy (large symmetry group) for persistent polynomials, as exhibited by explicit examples, e.g., determinants of sub-Hankel matrices. This reinforces their prehomogeneous and semi-invariant nature.

6. Applications in Quantum Information Theory and Tensor Rank

The development of persistent polynomials is intimately tied to questions in quantum information theory. Persistent symmetric tensors correspond to polynomial invariants whose rank properties persist under contractions (representing the loss of parties in a multipartite quantum state). This stability under reduction yields nontrivial lower bounds on tensor rank, generalizing minimal-rank entanglement structures such as the $W$-state to higher dimensions and degrees. The characterizations via polarization and the Hessian criterion establish rigorous bridges between tensor rank, complexity, and algebraic geometry.

7. Broader Implications and Open Directions

The algebraic and geometric framework developed for persistent polynomials motivates further exploration in several directions:

• Classification of persistent polynomials in larger $d$ and $n$.
• Refinement of the prehomogeneous and semi-invariant understanding in relation to the geometry of orbits and isotropy groups.
• Generalization of the link to homaloidal and Legendre-self-dual polynomials beyond cubics.
• Application to practical identification and classification of entanglement types in quantum information processing via persistent polynomial invariants.

The characterization of persistent polynomials as those for which the Hessian determinant is a pure power of a linear form provides a fundamental tool for their identification and analysis, unifying perspectives from algebraic geometry, invariant theory, and quantum information theory (Gharahi et al., 8 Oct 2025).