Nonunique nearest stable complex quadratics

Abstract

The closed set of complex monic quadratics with roots in the closed left half-plane is not prox-regular at z². For every fixed positive-definite real inner product on coefficient space, unstable quadratics with multiple attained global nearest stable quadratics occur arbitrarily close to z². The conclusion also holds in the full exact-degree quadratic coefficient space. A direct proof exhibits a limiting normal whose cubic support dominates a quartic squared displacement. The same obstruction disproves set prox-regularity of the whole abscissa epigraph. In contrast, the monic quadratic abscissa is function prox-regular at the origin relative to each finite limiting subgradient. Thus the historical question about local unique stable-polynomial projection has a negative answer for complex coefficients, while this example does not refute the separate function condition.

The idea in plain English

A polynomial is stable here when all its roots have nonpositive real part. Roots on the imaginary axis count as stable. One way to correct an unstable polynomial is to change its coefficients as little as possible until it becomes stable. The closest stable polynomial can be ambiguous: two different stable polynomials may be equally close.

This paper proves that such ties occur arbitrarily close to z² for complex quadratics. The conclusion holds for every fixed positive-definite real inner product used to measure coefficient changes, including metrics that mix the real and imaginary coordinates. Each neighborhood contains some unstable input with a tie; the theorem does not say that every nearby input has one.

The paper also separates two notions of local regularity. The stable set and the whole epigraph of the root-abscissa function have an obstruction to set prox-regularity. A different function condition still holds after fixing a finite limiting subgradient and allowing its local constants to depend on that subgradient.

The precise result

For the complex monic quadratic p(z)=z²+az+b, let α(p) be the maximum real part of a root and let S={(a,b)∈ℂ²:α(p)≤0}. Identify ℂ² with ℝ⁴ in the order (Re a, Im a, Re b, Im b). For each fixed real symmetric positive-definite coefficient matrix G, S is not prox-regular at (0,0), and every neighborhood of that point contains some unstable quadratic with at least two distinct attained global nearest points in S. The comparison is with all complex stable quadratics, including repeated and imaginary-axis roots.

The corresponding conclusion holds in the full exact-degree space S_full={(ℓ,a,b)∈ℂ³:ℓ≠0, α(ℓz²+az+b)≤0}, at (1,0,0), for every fixed real symmetric positive-definite coefficient metric on ℝ⁶. Both real leading-coefficient directions are included. The minimizers are attained global nearest points in the original exact-degree set, even though that set is not globally closed.

The entire abscissa epigraph {(c,r):α(c)≤r} fails set prox-regularity at height zero over the monic and full-degree base points, for every fixed positive-definite real metric on the respective coefficient/height space. The metric may mix height and coefficient coordinates. The limiting support covector is horizontal; its representing normal vector can tilt in a mixed metric.

Separately, the monic complex quadratic abscissa is function prox-regular at (0,0), relative to each fixed finite limiting subgradient v₀, in the real Euclidean coefficient metric. The local constants may depend on the chosen v₀. The function condition localizes subgradients, base points and function values; its epigraph normal is (v₀,−1), rather than the horizontal covector used for the set obstruction.

Why the result holds

Writing a=x+iy and b=u+iv gives the exact closed-stability chart x≥0, U=u+y²/4≥0 and V²≤x²U, where V=v−xy/2. Both nonnegativity conditions are needed at the singular faces. Along the stable path cₜ=(t²,0,t²,t³), smooth boundary normals converge to a limiting support covector n₀=(0,0,0,1). Its support is n₀·cₜ=t³, while the squared Euclidean displacement is 2t⁴+t⁶. The same cubic-versus-quartic obstruction persists in each fixed positive-definite metric.

The nearby normals are actual proximal normals: a sufficiently small normal step gives a local nearest point, and the triangle inequality excludes more distant competitors. Their limiting normal is not proximal. The classical Poliquin–Rockafellar–Thibault projection characterization then converts failure of set prox-regularity into nearby nonunique global projection. Closedness and a bounded minimizing-sequence argument supply monic attainment.

For full degree, the proof clips the stable set inside a closed metric ball of radius R that stays within ℓ≠0. The clipped set is compact. For a target less than R/3 from the base, points outside that ball are more than 2R/3 away, while the stable base is less than R/3 away. Thus the clipped and original global minimizers coincide. This compact localization supplies the full-degree attainment statement.

For the positive function result, the invertible discriminant-coordinate map (a,b)↦(a,a²−4b) transports lower supports in both directions. Fixing a finite subgradient bounds the scalar subgradients in a small neighborhood. A distance-to-ray estimate and the exact quadratic discriminant remainder then give the required local lower quadratic support.

Scope and limitations

The claims concern complex quadratics with closed-left-half-plane stability. They give no all-degree or real-only classification. For real monic quadratics the closed stability set is the convex orthant a≥0,b≥0, which has unique projection in every fixed positive-definite inner product.

Each metric is fixed and positive definite. No conclusion is asserted for arbitrary norms, singular or varying metrics, or a general transfer through nonlinear parameter-dependent metrics.

The nonuniqueness statement is existential in every neighborhood. It does not apply to every nearby unstable target, and it concerns global attained minimizers rather than competing local algorithmic minima. No explicit unstable input together with its two minimizing stable polynomials, or algorithm computing such a pair, is supplied.

The full exact-degree set excludes ℓ=0 and is not globally closed. Its global-attainment claim depends on the compact localization argument; treating the entire set as closed would not justify that conclusion.

The positive function claim is restricted to the monic abscissa and real Euclidean coefficient metric, relative to each fixed finite limiting subgradient. It does not give constants uniform over all finite subgradients, a full-degree function theorem, or a function theorem for every mixed coefficient/height metric. Whole-epigraph set failure does not refute this separate function condition.

Poliquin and Rockafellar’s Theorem 3.1 is a prox-regularity composition result under hypotheses; it does not itself state unconditional exact subdifferential equality. The manuscript’s explicit two-sided diffeomorphism and lower-support argument independently supplies the equality used here.

Finite exact rational checks and high-precision Decimal controls supplement the analytic proofs. Decimal controls are not interval-certified bounds, and finite computations do not establish the limiting-normal, universal-metric or global projection conclusions by themselves.

The short 2005 Oberwolfach question does not specify coefficient field, normalization or metric. The comparison here uses the explicit complex monic convention in Lewis’s 2003 Section 8; no unqualified real-only answer is asserted.

The original Cross, Moses–Liu and Bottema bodies were not inspected for the detailed comparisons made in the manuscript. Those exact example and nonuniqueness comparisons are not asserted here. Earlier coefficient-approximation work and related stability-boundary geometry receive credit without an exhaustive absence-of-overlap claim.

No historical firstness, exhaustive novelty clearance or continued worldwide openness is claimed.

Status and AI use

AI tools were used extensively in research, derivation, computational checks, literature comparison and writing, including adversarial AI reviews. These reviews are not independent human or journal peer review. This is an unrefereed preprint without formal proof-assistant certification. The named author is responsible for the claims and citations.

The manuscript is dated 11 October 2026 and labels itself a research note and unrefereed preprint without printing a version number. Zenodo identifies this edition as version 1.0 and records creation on 11 October 2026 UTC. These fields do not establish first priority.

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Cite this paper

Alec Kriebel. (2026). Nonunique nearest stable complex quadratics (Zenodo version 1.0) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.23306332

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Background and prior work

• Lewis — The mathematics of eigenvalue optimization (2003): Section 8, printed pp.166–168, specifies complex monic coefficients, a real Euclidean coefficient metric and closed-half-plane stability before asking about set prox-regularity. Quadratics correspond to n=1 in its z^(n+1)+p notation.
• Lewis, joint with Burke and Overton — Local Structure and Algorithms in Nonsmooth Optimization (2005): The original contribution occupies printed pp.104–106; Question 1 on p.106 asks about nearby unique nearest stable polynomials. Its short wording leaves field, normalization and metric unspecified, so the complex convention is qualified through the earlier Lewis source.
• Poliquin, Rockafellar and Thibault — Local differentiability of distance functions (2000): Definition 1.1, Proposition 1.2 and Theorem 1.3(a),(k) supply the classical set prox-regularity and global projection characterization. Closedness is essential, and the additional local weak-closedness condition is automatic in finite dimensions.
• Poliquin and Rockafellar — A calculus of prox-regularity (2010): Definition 1.1 localizes finite subgradients for the function condition; p.205 gives the epigraph correspondence at (v,−1). Theorem 3.1 is a conditional composition result. The present proof supplies its own exact diffeomorphism transport.
• Burke and Overton — Variational analysis of the abscissa mapping for polynomials (2001): Classical complex polynomial-abscissa conventions and first-order variational analysis. Root and normal calculus are prior tools.
• Burke, Lewis and Overton — Variational analysis of functions of the roots of polynomials (2005): Earlier variational analysis in complex exact-degree polynomial coordinates; this background is credited separately from the present local projection argument.
• Kirillov and Overton — Robust stability at the Swallowtail singularity (2013): Related real-quartic Whitney-umbrella stability geometry, including credit to earlier work of Bottema. The differing normalization and open Hurwitz domain are not identified with the present complex quadratic global-projection problem.
• Orban de Xivry — Nearest Stable System (2013): Selected sections of Chapters 2 and 4 document earlier complex monic coefficient approximation, marginal-stability closure, algorithmic methods and weighted objectives. These bounded readings establish prior work without an exhaustive theorem-overlap audit.
• Moses and Liu — Determining the closest stable polynomial to an unstable one (1991): Earlier nearest-stable-polynomial literature cited by the manuscript. The original body was not inspected, so no precise original nonuniqueness statement or example comparison is asserted here.