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A Sixth-Order Hurwitz Interval Polynomial Family Without Degree-Five Weak Strictly Positive Real Numerator

Primarily AI-generated textHuman understanding: some partsmath.OC — Optimization and Control

Contributed by Tiancheng Xu ↗, Zhengtai Xiong, Wensheng Yu

Version 1 / Nov 05, 2026 / CC BY 4.0

Abstract

We construct an explicit sixth-order Hurwitz-stable interval polynomial family with independent closed coefficient intervals that admits a common strictly positive real (SPR) numerator of degree six, yet admits no common weak strictly positive real (WSPR) numerator of degree five. More strongly, no nonzero real polynomial of degree at most five makes the real part of the associated transfer functions nonnegative across the family at all real frequencies. This counterexample shows that, for sixth-order interval polynomial families, robust Hurwitz stability does not guarantee the existence of a strictly proper common SPR or WSPR numerator of degree five, and equal numerator degree six is necessary in general.

Provenance statement

OpenAI’s GPT-6 Astra was used for calculations, proof development, and editorial assistance during the preparation of this work. AI-generated suggestions were checked against the mathematical arguments and the accompanying formal development. The main counterexample theorem and its supporting results have been formalized in Lean 4, with proofs checked by the Lean kernel. The formalization is available at https://github.com/dgjszy/sextic-spr-degree-necessity. LaTeX was used to prepare the manuscript. The authors take responsibility for the final manuscript and its mathematical claims.

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References

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Version history

  1. v1 · 10 pagesSubmitted by Tiancheng XuInitial depositCurrentNov 05, 2026