Extend variables without a known determinantal representation

Construct meaningful real-zero variable extensions of a polynomial without presupposing a determinantal representation, in a manner that can enlarge the relaxation and potentially recover a determinantal representation satisfying the generalized Lax conjecture.

Background

The dissertation observes that extending a determinantal polynomial by adding variables corresponding to a basis of the symmetric matrix space can produce the type of cofactor and rigidly convex-set containment required by the generalized Lax conjecture. However, this construction assumes that a determinantal representation is already available.

A general method that starts only from the coefficients of a real-zero polynomial would make the relaxation substantially more powerful and could provide a route toward the generalized Lax conjecture. The text explains that the relevant trace products might be recoverable from L-form data, but leaves the construction unresolved.

References

Now, it is not clear how to extend the variables without having already previously a determinantal representation.

— A Method for Establishing Asymptotically Accurate Bounds for Extremal Roots of Eulerian Polynomials Using Polynomial Stability Preservers  (2503.04628 - Nevado, 6 Mar 2025) in Remark “Extending the monomials in terms of traces,” subsection “Extending the matrices”