Determine higher-degree L-form values from polynomial data

Determine how to define the value of the L-form on higher-degree monomials, such as x_1^2x_2^2, so that it captures the relevant noncommutative trace information of a determinantal representation without requiring access to that representation.

Background

The relaxation uses L-form values through degree three, where traces of products of the representing matrices can be recovered in a sufficiently simple way. At higher degree, different orderings of noncommuting matrix products can have different traces even though they correspond to the same commutative monomial.

Resolving this issue would be necessary for enlarging the moment mold and incorporating higher-degree coefficient information into stronger relaxations. The dissertation identifies a likely connection with noncommutative liftings but does not develop such a theory.

References

This is no longer true for degree higher than $4$ because for example $\tr(A_{1}A_{2}A_{1}A_{2})\neq\tr(A_{1}A_{2}A_{2}A_{1})$ in general and therefore, without access to the determinantal representation, it is not clear what should the value of $L_{p}(x{2}{1}x{2}{2})$ be in order to capture that product of traces into the matrix defining the relaxation.

— 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 “L-form behaviour and noncommutativy problems for degree higher than three,” Section “Extending the matrices”