Construct nontrivial multivariate extensions admitting determinantal representations

Construct, for real-zero polynomials p of degree d, instances of coefficients a_0,\ldots,a_d such that the polynomial \sum_{k=0}^{d}a_k y^k p^{(k)} in the variables (x,y) admits a determinantal representation, beyond the already established exponential-coefficient extension.

Background

The dissertation considers extensions formed from Renegar derivatives, including the established polynomial \hat p=\sum_{k=0}{d}yk p{(k)}/k!, which has a determinantal representation under suitable hypotheses. The authors seek further coefficient choices that yield determinantal representations and could provide more informative variable extensions.

Such extensions are relevant to improving the polynomial relaxation and potentially advancing the generalized Lax conjecture. The text reports that no additional nontrivial multivariate extensions of the desired type were found beyond the particular construction already presented.

References

Sadly, we could not find more nontrivial multivariate extensions than the one presented in Proposition \ref{intermedia} (and others obtained in a similar way by considering determinantal representations of size higher than the degree $d$ of $p$, which always exist trivially whenever $p$ has a determinantal representation) for the results in .

— A Method for Establishing Asymptotically Accurate Bounds for Extremal Roots of Eulerian Polynomials Using Polynomial Stability Preservers  (2503.04628 - Nevado, 6 Mar 2025) in Problem “Other expansions and extensions,” following Remark “Fixing problems with homogenizing degree”