Triviality of Frobenius radical elements in the generic axial algebra U_λ

Determine whether every generalized polynomial f lying in the radical of the Frobenius form on the generic axial algebra U_λ must be zero; equivalently, decide whether there exist nonzero generalized polynomials f such that (f, X) is an axial Frobenius identity of Jordan type λ for all specializations, which would imply the presence of nontrivial universal Frobenius identities across nonsingular axial algebras of Jordan type λ.

Background

The paper constructs a generic nonsingular axial algebra of Jordan type λ and discusses the notion of the radical both in an intrinsic (Baer-type) sense and in terms of a Frobenius form. It notes that a generalized polynomial f lies in the radical of the Frobenius form precisely when (f, X) is an axial Frobenius identity of Jordan type λ.

Resolving whether such an f must be trivial would clarify whether the generic Frobenius form is nondegenerate and whether there exist nontrivial universal Frobenius identities that hold across all nonsingular axial algebras of Jordan type λ. The authors point out that if f were nonzero, it would yield idempotental Frobenius identities in all nonsingular axial algebras of Jordan type λ, but they do not see how this would lead to a contradiction.