The generalized Lax conjecture is true for topological reasons related to compactness, convexity and determinantal deformations of increasing products of pointwise approximating linear forms

TL;DR

This paper proves the generalized Lax conjecture using topological methods, showing that determinants of symmetric linear pencils can represent rigidly convex sets of RZ polynomials, with perturbation techniques ensuring the sets are preserved.

Contribution

It introduces a topological approach to prove the generalized Lax conjecture, demonstrating the existence of determinantal representations for RZ polynomials via perturbation methods.

Findings

Determinants of symmetric linear pencils can express rigidly convex sets of RZ polynomials.

A topological perturbation argument ensures the preservation of convex sets in representations.

The proof confirms the generalized Lax conjecture in a broad topological context.

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.