Continuity of the solution map for hyperbolic polynomials

Problem Statement

The paper addresses the regularity and continuity properties of the solution map for hyperbolic polynomials. A monic real polynomial of degree $d$ is termed hyperbolic if all its roots are real. The central object of study is the "solution map" $S$, which maps the coefficients of such a polynomial to its increasingly ordered roots.

While Bronshtein's theorem establishes that for a family of hyperbolic polynomials with coefficients in the class $C^{d-1,1}$ (functions with $(d-1)$-th derivatives that are Lipschitz continuous), the roots admit a continuous parameterization that is locally Lipschitz ($C^{0,1}$), the continuity of the solution map itself with respect to the topology of the coefficient space remained an open question. Specifically, the authors investigate whether the map $S: C^{d-1,1} \to C^{0,1}$ is continuous. As demonstrated by counterexamples in the paper, this map is not continuous when the target space is equipped with the standard $C^{0,1}$ (Lipschitz) topology. The paper seeks to determine if continuity can be recovered by relaxing the topology on the target space (e.g., to Sobolev spaces) or by strengthening the regularity of the source space (e.g., to $C^d$).

Methodology

The authors employ a combination of real analysis, the theory of hyperbolic polynomials, and functional analysis. The core methodology involves:

1. Tschirnhausen Transformation: The polynomials are transformed into Tschirnhausen form (where the coefficient of $Z^{d-1}$ is zero) to simplify the analysis of the roots. This allows for the use of scaling arguments and the study of the space of polynomials with a fixed second coefficient.
2. Splitting Principle: A key technique involves the local splitting of hyperbolic polynomials into factors of lower degree. Using the inverse function theorem and the structure of the discriminant locus, the authors show that near points where roots are distinct or have specific multiplicity structures, the polynomial can be decomposed into hyperbolic factors with analytic dependence on the coefficients.
3. Single-Point Bronshtein Theorem: The authors develop a refined version of Bronshtein's theorem at a single point (Theorem 4.7). This provides explicit bounds on the derivative of a root at a point $x_0$ in terms of the coefficients and their derivatives, provided $x_0$ is not a "reclusive point" (a specific type of isolated point in the zero set of the coefficients).
4. Dominated Convergence: To prove the convergence of the solution map in Sobolev norms, the authors establish that the sequence of derivatives of the roots is uniformly bounded (via Bronshtein's theorem) and converges pointwise almost everywhere. The Dominated Convergence Theorem is then applied to establish convergence in $W^{1,q}$ for $1 \le q < \infty$.
5. Induction on Degree: The proofs for the convergence of derivatives rely on an induction argument on the degree of the polynomial, utilizing the splitting principle to reduce the problem to polynomials of lower degree.

Key Contributions and Results

The primary contribution is the proof of the continuity of the solution map under specific topological constraints:

• Main Theorem (Theorem 1.1): The solution map $S$ from hyperbolic polynomials of degree $d$ with $C^d$ coefficients to their increasingly ordered roots is continuous when the source space carries the natural Fréchet topology of $C^d$ and the target space is equipped with the trace topology of the Sobolev space $W^{1,q}_{loc}$ for all $1 \le q < \infty$.
• Hölder Continuity (Corollary 1.2): As a consequence, the map is also continuous into the Hölder space $C^{0,\alpha}$ for all $0 < \alpha < 1$, but fails to be continuous for $\alpha = 1$ (the Lipschitz case).
• Technical Core (Theorem 1.3): The paper establishes that if a sequence of coefficients $a_n$ converges to $a$ in $C^d$, then the sequence of roots $S(a_n)$ converges to $S(a)$ in $W^{1,q}_{loc}$ for any $1 \le q < \infty$. This relies on the pointwise convergence of the derivatives almost everywhere, which is proven using the single-point Bronshtein variant.
• Applications to Matrices and Singular Values:
  • Hermitian Matrices: The results are applied to the eigenvalues of Hermitian matrices. The map taking a $C^d$ curve of Hermitian matrices to its ordered eigenvalues is shown to be continuous into $W^{1,q}_{loc}$ and $C^{0,\alpha}$.
  • Singular Values: An analogous result is obtained for the singular values of $C^{2d}$ families of complex matrices.
• Geometric Consequences:
  • Surface Area: The paper proves that the surface area of the graph of each root converges locally.
  • Zero Sets: The area of the zero sets of hyperbolic polynomials exhibits local lower semicontinuity. Specifically, if $a_n \to a$ in $C^d$, the Hausdorff measure of the zero sets satisfies $\liminf_{n \to \infty} \mathcal{H}^m(Z_n) \ge \mathcal{H}^m(Z)$.
• Approximation: The authors show that any hyperbolic polynomial with $C^d$ coefficients can be approximated by a sequence of hyperbolic polynomials with simple roots (distinct roots everywhere), such that the roots of the approximating sequence converge to the roots of the original in the $W^{1,q}_{loc}$ topology.

Significance and Claims

The paper claims to resolve Open Problem 3.8 from a previous survey by the authors ([PR25]). The significance lies in clarifying the optimal regularity and continuity of the solution map for hyperbolic polynomials, a problem with deep roots in the perturbation theory of linear operators and partial differential equations.

The authors emphasize that while Bronshtein's theorem guarantees the existence of Lipschitz roots for $C^{d-1,1}$ coefficients, the continuity of the map assigning these roots to the coefficients requires higher regularity ($C^d$) on the source and a weaker topology ($W^{1,q}$ or $C^{0,\alpha}$) on the target. The failure of continuity in the $C^{0,1}$ target topology is explicitly demonstrated, highlighting the sharpness of the results.

The paper also notes that the results for hyperbolic polynomials are stronger than those for general complex polynomials (treated in [PR24]), where the solution map is only continuous into $W^{1,q}$ for $q < d/(d-1)$. For hyperbolic polynomials, continuity holds for all $q < \infty$.

The authors remain modest regarding the $C^{d-1,1}$ case, posing it as an open question (Question 1.7) whether the continuity results extend to the solution map on $C^{d-1,1}(U, \text{Hyp}(d))$. Similarly, for Hermitian matrices, they ask if the regularity requirement can be lowered from $C^d$ to $C^1$ (Question 1.11), noting that they can only affirmatively answer this for the case $d=2$.