Symmetric Persistent Tensors and their Hessian
Masoud Gharahi, Giorgio Ottaviani
TL;DR
The paper identifies a precise algebraic criterion linking symmetric persistent tensors to Hessian factorization: a homogeneous $f\in \mathrm{Sym}^n\mathbb{C}^d$ is persistent if and only if ${\rm Hess}(f)$ is a power of a linear form, yielding a practical detector via a chain of implications and an explicit multi-derivative characterization. It proves the key equivalences $(a)\Rightarrow(b)\Leftrightarrow(c)\Rightarrow(d)$, with $(b)\Leftrightarrow(c)$ giving an effective test and $(a)\Rightarrow(a')$ ensuring sufficiency, while noting $(b)\Rightarrow(a)$ holds in special cases ($n=3$ or $d\le4$). The work provides complete small-dimension normal forms, e.g., for $d\le3$ and for $(d,n)=(4,3)$, and situates persistence in prehomogeneous geometry, linking to semi-invariants, homaloidal polynomials, and Legendre transforms. It further develops constructive families of persistent polynomials (e.g., weight $d-1$ isobaric forms and sub-Hankel determinants $f_{(d)}$), explores Perazzo-type structures, and demonstrates that all persistent cubics are homaloidal, indicating deep connections between tensor rank stability and classical algebraic geometry with potential implications for quantum information theory.
Abstract
Persistent tensors, introduced in [Quantum 8 (2024), 1238], and inspired by quantum information theory, form a recursively defined class of tensors that remain stable under the substitution method and thereby yield nontrivial lower bounds on tensor rank. In this work, we investigate the symmetric case-namely, symmetric persistent tensors, or equivalently, persistent polynomials. We establish that a symmetric tensor in $\mathrm{Sym}^n \mathbb{C}^d$ is persistent if the determinant of its Hessian equals the $d(n-2)$-th power of a nonzero linear form. The converse is verified for cubic tensors ($n=3$) or for $d \leq 3$, by leveraging classical results of B. Segre. Moreover, we demonstrate that the Hessian of a symmetric persistent tensor factors as the $d$-th power of a form of degree $(n-2)$. Our main results provide an explicit necessary and sufficient criterion for persistence, thereby offering an effective algebraic characterization of this class of tensors. Beyond characterization, we present normal forms in small dimensions, place persistent polynomials within prehomogeneous geometry, and connect them with semi-invariants, homaloidal polynomials, and Legendre transforms. Particularly, we prove that all persistent cubics are homaloidal.