The generalized Lax conjecture is true for topological reasons related to compactness, convexity and determinantal deformations of increasing products of pointwise approximating linear forms
Alejandro González Nevado
TL;DR
The paper tackles the generalized Lax conjecture by developing a topological framework in which rigidly convex sets of real-zero polynomials are realized as spectrahedral representations through determinants of large symmetric linear pencils. It leverages compactness, perturbations, and smooth deformations of tangent hyperplanes to build determinantal representations that preserve the original rigidly convex set, bypassing dimensionality barriers that plague purely algebraic approaches. A key construction uses finite tangent coverings of the real zero set and gluing via block-diagonal assembly to obtain a determinant representation for a product $qp$ of the target $RZ$ polynomial $p$, with the rigid convex set preserved along the deformation. The result establishes the generalized Lax conjecture in a topological sense and clarifies why effective explicit representations are difficult, while offering a robust pathway for future algorithmic and algebraic refinements.
Abstract
We develop a topological approach to prove the generalized Lax conjecture using the fact that determinants of sufficiently big symmetric linear pencils are able to express the rigidly convex sets of RZ polynomials of any degree $d$. Monicity of the representation is assessed through a topological argument that allows us to perturbate a sufficiently close linear approximation into a suitable nice determinantal multiple of the initial RZ polynomial with the same rigidly convex set. The perturbation can be smoothly performed. This fact is what will allow us to determine that the multiple obtained respects the initial rigidly convex sets. This argument provides thus a full proof of the generalized Lax conjecture. However, an effective proof providing the representation in nice terms seems far from reachable at this moment.