Legendre transform pattern for sub-Hankel determinant polynomials

Determine the explicit form of the multiplicative Legendre transform (f_{(d)})_* for the family of persistent polynomials f_{(d)} = det M_{(d)} (generic sub-Hankel determinants) in dimensions d ≥ 5, and establish whether the projective equivalences observed in lower dimensions—such as (f_{(4)})_* being projectively equivalent to f_{(4)}^2/x_0^3 and (f_{(4)} x_0)_* being projectively equivalent to f_{(4)}^3/x_0^5—extend as a uniform pattern to higher d.

Background

The multiplicative Legendre transform provides the inverse to the polar map for homaloidal polynomials. The authors compute this transform for specific persistent examples, including f_{(3)}, f_{(4)}, and f_{(4)} x_0, and observe a recurring projective-equivalence pattern in how the transform expresses as a rational function of powers of f and x_0.

They suggest that a similar pattern may hold for the broader family f_{(d)} obtained from sub-Hankel determinants, but the general case remains unresolved. Establishing an explicit formula or uniform equivalence would deepen understanding of the interplay between persistence, homaloidality, and prehomogeneous geometry.