Conjecture: Persistent polynomials are homaloidal

Prove that every symmetric persistent tensor f (equivalently, every persistent homogeneous polynomial on C^d) is homaloidal, i.e., its polar map ∇f: P^{d−1} ↠ (P^{d−1})^∨ is birational.

Background

The paper connects persistence to the behavior of the Hessian and explores relationships with homaloidal polynomials via the polar map. It is shown (following Mammana) that cubic polynomials whose Hessian is a power of a linear form are homaloidal; consequently, all persistent cubics are homaloidal.

Based on extensive examples and structural properties, the authors propose a broader conjecture that persistence implies homaloidality in all degrees and dimensions. They note that the converse is false (e.g., f = x_0 x_1 x_2 is homaloidal but not persistent), highlighting the one-directional nature of the conjecture.