Necessity of the linear-form power condition for persistence

Ascertain whether condition (a) in Theorem 1 is necessary in general: prove or refute that, for every persistent homogeneous polynomial f ∈ Sym^n C^d with n > 2, there exists a nonzero linear form ℓ such that (f) = ℓ^{d(n−2)}. The equivalence holds for n = 3 and for d ≤ 4, but the general case across all n and d remains undetermined.

Background

The main theorem presents a chain of implications connecting persistence to factorization properties of the Hessian: (a) ⇒ (b) ⇔ (c) ⇒ (d). Here, (a) asserts that (f) equals the d(n−2)-th power of a nonzero linear form ℓ, (b) is persistence, (c) is a polarized Hessian factorization criterion, and (d) is a weaker necessary condition stating that (f) is a d-th power of a degree-(n−2) form.

The authors prove sufficiency of (a) and necessity of (d), with (b) ⇔ (c) providing an algorithmic test for persistence. However, whether the strong factorization in (a) is also necessary for persistence is unknown in general. They verify necessity for special cases: cubic tensors (n = 3) and small dimensions d ≤ 4.