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Continuity of the solution map for hyperbolic polynomials

Adam Parusiński, Armin Rainer

TL;DR

The paper studies the continuity of the coefficient-to-root map for monic hyperbolic polynomials, proving that when coefficients depend in C^d, the increasingly ordered roots vary continuously in the Sobolev spaces C^{0,1}_q for all 1 ≤ q < ∞ (but not for q = ∞). Central to the argument is a main technical result (Theorem mainhyp) established via a single-point Bronshtein-type estimate and a simultaneous polynomial splitting, which yields convergence of root derivatives in L^q and strong C^0 convergence on compact intervals. The authors develop a robust framework: a Tschirnhausen normalization, splitting, universal splitting data, and admissible data, enabling an induction on degree and a multiparameter extension. Consequences include continuity and area-continuity properties for root graphs, lower semicontinuity of zero-set areas, and practical perturbation results for eigenvalues of Hermitian matrices and singular values. The findings have significance for stability analysis in spectral theory and provide refined tools for hyperbolic polynomial perturbations with potential broad applicability in numerical analysis and PDE perturbation theory.

Abstract

Hyperbolic polynomials are monic real-rooted polynomials. By Bronshtein's theorem, the increasingly ordered roots of a hyperbolic polynomial of degree $d$ with $C^{d-1,1}$ coefficients are locally Lipschitz and the solution map "coefficients-to-roots" is bounded. We prove continuity of this solution map from hyperbolic polynomials of degree $d$ with $C^d$ coefficients to their increasingly ordered roots with respect to the $C^d$ structure on the source space and the Sobolev $W^{1,q}$ structure, for all $1 \le q<\infty$, on the target space. Continuity fails for $q=\infty$. As a consequence, we obtain continuity of the local surface area of the roots as well as local lower semicontinuity of the area of the zero sets of hyperbolic polynomials. We also discuss applications for the eigenvalues of Hermitian matrices and singular values.

Continuity of the solution map for hyperbolic polynomials

TL;DR

The paper studies the continuity of the coefficient-to-root map for monic hyperbolic polynomials, proving that when coefficients depend in C^d, the increasingly ordered roots vary continuously in the Sobolev spaces C^{0,1}_q for all 1 ≤ q < ∞ (but not for q = ∞). Central to the argument is a main technical result (Theorem mainhyp) established via a single-point Bronshtein-type estimate and a simultaneous polynomial splitting, which yields convergence of root derivatives in L^q and strong C^0 convergence on compact intervals. The authors develop a robust framework: a Tschirnhausen normalization, splitting, universal splitting data, and admissible data, enabling an induction on degree and a multiparameter extension. Consequences include continuity and area-continuity properties for root graphs, lower semicontinuity of zero-set areas, and practical perturbation results for eigenvalues of Hermitian matrices and singular values. The findings have significance for stability analysis in spectral theory and provide refined tools for hyperbolic polynomial perturbations with potential broad applicability in numerical analysis and PDE perturbation theory.

Abstract

Hyperbolic polynomials are monic real-rooted polynomials. By Bronshtein's theorem, the increasingly ordered roots of a hyperbolic polynomial of degree with coefficients are locally Lipschitz and the solution map "coefficients-to-roots" is bounded. We prove continuity of this solution map from hyperbolic polynomials of degree with coefficients to their increasingly ordered roots with respect to the structure on the source space and the Sobolev structure, for all , on the target space. Continuity fails for . As a consequence, we obtain continuity of the local surface area of the roots as well as local lower semicontinuity of the area of the zero sets of hyperbolic polynomials. We also discuss applications for the eigenvalues of Hermitian matrices and singular values.
Paper Structure (47 sections, 46 theorems, 225 equations)

This paper contains 47 sections, 46 theorems, 225 equations.

Table of Contents

  1. Introduction
  2. Hyperbolic polynomials and the solution map
  3. The main results
  4. Applications
  5. Relation to the results for general polynomials
  6. Continuity of the area of the solution map
  7. Approximation by hyperbolic polynomials with simple roots
  8. Perturbation theory for Hermitian matrices
  9. Singular values
  10. On the optimality of the results
  11. Structure of the paper
  12. Notation
  13. Function spaces
  14. Hölder--Lipschitz spaces
  15. Lebesgue spaces
  16. ...and 32 more sections

Key Result

Theorem 1.1

[t]thm:solmap Let $U \subseteq \mathbb{R}^m$ be open. The solution map is continuous, for all $1\le q<\infty$.

Theorems & Definitions (94)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • proof
  • Theorem 1.6
  • Remark 1.8
  • Corollary 1.9
  • Theorem 1.10
  • Example 1.12
  • ...and 84 more