[
  {
    "id": "ARR-2026-2XFZW9FFEA8SXTXF-v1",
    "type": "article",
    "title": "Finite convex interpolation for polynomial Lyapunov existence",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          10,
          20
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-2XFZW9FFEA8SXTXF",
    "version": "v1",
    "genre": "Research preprint",
    "URL": "https://airr.science/papers/ARR-2026-2XFZW9FFEA8SXTXF/versions/v1/",
    "note": "Research preprint; ARR-2026-2XFZW9FFEA8SXTXF v1. Not peer reviewed by AIRR. Working paper; not admitted to the AIRR accepted collection.",
    "abstract": "We characterize polynomial Lyapunov existence over all degrees for planar homogeneous polynomial fields with nonvanishing angular velocity. A moment threshold separates strict polynomial certificates from the absence of local analytic weak certificates in the stable case. Classical confluent Nevanlinna-Pick interpolation gives a finite convex semialgebraic formulation, with exact boundary decision for algebraic coefficients. For each odd field degree, the certificate region is open and semialgebraic, and compact admissible families have a computable common certificate degree. For rational fields in dense binary encoding, we prove polynomial-time additive approximation of the threshold and polynomial-time decision under a supplied positive gap promise. Explicit inner and outer radii permit a classical weak convex optimization reduction; a uniform functional perturbation argument handles algebraic poles without constructing their common number field. This approximates a value and does not bound the output size of a Lyapunov polynomial polynomially. Jackson bounds and a constructive sufficient-degree estimate complement the finite formulation. An exact cubic example has an algebraic threshold and minimum Lyapunov degree eight at one rational parameter. Beyond the rotating class, a boundary-factor obstruction produces a full twelve-dimensional rational box of stable quintic fields with sign-changing angular velocity and no local analytic weak certificate. One explicit rational C1 Lyapunov function works for the entire box. The latter result is a sufficient obstruction, not a general classification with angular zeros. General certified solver implementations are not supplied."
  }
]