Continuity of the solution map for hyperbolic polynomials

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.

💡 Research Summary

The paper investigates the continuity properties of the solution map that sends the coefficients of a hyperbolic (i.e., monic real‑rooted) polynomial of degree d to its increasingly ordered real roots. Classical results due to Bronshtein (1979, 1980) guarantee that if the coefficient functions belong to the Hölder class C^{d‑1,1}, then the roots can be chosen as locally Lipschitz functions of the parameters; consequently the solution map S : C^{d‑1,1}(U, Hyp(d)) → C^{0,1}(U,ℝ^{d}) is bounded. However, this map is not continuous when the target space carries the full C^{0,1} topology, as shown by a simple counter‑example (Example 1.12).

The authors’ main contribution is to show that continuity is recovered if the target space is equipped with a weaker Sobolev topology. Specifically, for any open set U⊂ℝ^{m} and any 1 ≤ q < ∞, the solution map S : C^{d}(U, Hyp(d)) → C^{0,1}{q}(U,ℝ^{d}) is continuous, where C^{0,1}{q}(U,ℝ^{d}) denotes the space C^{0,1}(U,ℝ^{d}) endowed with the trace topology induced by the inclusion into the Sobolev space W^{1,q}_{loc}(U,ℝ^{d}). Theorem 1.1 establishes this result, while Corollary 1.2 shows that the map is also continuous into the Hölder spaces C^{0,α} for every 0 < α < 1, but fails for α = 1.

The technical heart of the paper is Theorem 1.3, which treats the one‑dimensional parameter case. If a sequence of coefficient curves a_n converges to a in C^{d} on every compact subinterval of an open interval I, then the corresponding root curves S(a_n) form a bounded set in C^{0,1}(I,ℝ^{d}) and converge to S(a) in W^{1,q}(I_0,ℝ^{d}) for every compact I_0⊂I and every 1 ≤ q < ∞. The proof relies on a pointwise version of Bronshtein’s theorem (Theorem 4.7) and the dominated convergence theorem; the domination is supplied by the original Bronshtein Lipschitz bound. A refinement (Theorem 5.1) shows that the convergence hypothesis can be weakened to C^{p} where p is the maximal multiplicity of the roots of P_a.

Several important consequences are derived:

• Surface‑area continuity (Section 7.2): The Jacobian determinant of the solution map converges in L^{q}_{loc}, which, via the area formula, implies that the surface area of each root graph varies continuously under C^{d} perturbations of the coefficients. Consequently, the m‑dimensional Hausdorff measure of the zero set Z = {(x,y) ∈ U×ℝ | P_a(x)(y)=0} is lower semicontinuous (Corollary 1.9).

• Approximation by simple‑root hyperbolic polynomials (Section 7.3): Every hyperbolic polynomial family can be approximated in C^{d} by families whose roots are simple everywhere. The approximating families inherit the continuity properties of the solution map, and the associated zero‑set measures satisfy the same lower‑semicontinuity.

• Hermitian eigenvalues (Section 7.4): Ordering the eigenvalues of a Hermitian matrix yields a continuous map λ↑: Herm(d)→ℝ^{d}. By Weyl’s perturbation theorem, the induced map E : C^{0,1}(U, Herm(d)) → C^{0,1}(U,ℝ^{d}) is bounded. Theorem 1.10 shows that if the matrix‑valued coefficients belong to C^{d}, then E is continuous into C^{0,1}_{q} for all 1 ≤ q < ∞, and into C^{0,α} for every 0 < α < 1. The map fails to be continuous in the full C^{0,1} topology (Example 7.13). For d = 2, continuity already holds under a C^{1} assumption (Proposition 7.14).

• Singular values (Section 7.5): An analogous result holds for the ordered singular values of a family of complex matrices A(x) with d ≤ D. If the entries of A are C^{2d}, then the singular‑value map is continuous into C^{0,1}_{q} for all q < ∞.

The paper also discusses optimality. Example 1.12 demonstrates that the solution map is not continuous when the target space carries the full C^{0,1} topology (i.e., q = ∞). Moreover, the authors raise open questions about whether the continuity results could be extended to the weaker coefficient regularity C^{d‑1,1} or even C^{1} in the Hermitian case, and they point out that additional structural assumptions (e.g., the discriminant set having measure zero) would be needed for such extensions.

Structurally, the paper proceeds as follows: Section 2 defines the functional spaces used (Hölder, Sobolev, trace topologies). Section 3 reviews basic properties of hyperbolic polynomials and the coefficient‑to‑root map. Section 4 presents Bronshtein’s theorem and a pointwise refinement. Sections 5 and 6 contain the proofs of the main technical theorem and the main continuity results. Section 7 explores applications (surface area, zero‑set semicontinuity, eigenvalues, singular values) and discusses optimality. Finally, Section 8 treats the case of bounded root multiplicities, showing that the required regularity can be lowered accordingly.

In summary, the authors establish that the solution map for hyperbolic polynomials is continuous when the source space is equipped with C^{d} regularity and the target space is measured in Sobolev W^{1,q} norms (1 ≤ q < ∞). This bridges a gap between the classical Lipschitz regularity guaranteed by Bronshtein’s theorem and the stronger continuity needed for geometric and spectral applications, providing a robust framework that applies to eigenvalue and singular‑value perturbation theory as well as to the analysis of zero‑set measures.