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Real-valued Jordan curves and nonreal limiting spectra of banded Toeplitz matrices

by F. Rohner and C. Thalhammer

(Report number 2026-34)

Abstract
We construct Laurent polynomials that are real-valued on a Jordan curve surrounding the origin but whose banded Toeplitz matrices have nonreal limiting spectrum. This refutes the weakest surviving form of a conjecture of Shapiro and Štampach.

Keywords: Banded Toeplitz matrices, limiting spectrum, Laurent polynomials, Jordan curves, equal-modulus roots

@Techreport{RT26_1189,
  author = {F. Rohner and C. Thalhammer},
  title = {Real-valued Jordan curves and nonreal limiting spectra of banded Toeplitz matrices},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2026-34},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2026/2026-34.pdf },
  year = {2026}
}

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