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PreprintCC BY-NC-ND 4.0arXiv

Real rational Herglotz-Nevanlinna functions in two variables and a class of extremals

  • Ludvig Fagrell
  • Annemarie Luger
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Abstract

We first revisit the Nevanlinna condition for representing measures of Herglotz-Nevanlinna functions in two variables and give a general operator-theoretic criterion for extremality. We then derive an explicit formula for the representing measure of a real rational Herglotz-Nevanlinna function in two variables, and combine it with the extremality criterion to obtain a family of extreme elements in the full cone of Nevanlinna measures. The family includes examples with reducible denominators, and these yield counterexamples to the irreducibility assertion in a conjecture of Knese. We then characterize reducibility within this family both algebraically and geometrically.

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Cite

BibTeXRIS
BibTeX
@misc{fagrell2026real,
  title = {{Real rational Herglotz-Nevanlinna functions in two variables and a class of extremals}},
  author = {Ludvig Fagrell and Annemarie Luger},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.02476},
  eprint = {2610.02476},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.02476},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Real rational Herglotz-Nevanlinna functions in two variables and a class of extremals
AU  - Ludvig Fagrell
AU  - Annemarie Luger
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.02476
UR  - https://arxiv.org/abs/2610.02476
AB  - We first revisit the Nevanlinna condition for representing measures of Herglotz-Nevanlinna functions in two variables and give a general operator-theoretic criterion for extremality. We then derive an explicit formula for the representing measure of a real rational Herglotz-Nevanlinna function in two variables, and combine it with the extremality criterion to obtain a family of extreme elements in the full cone of Nevanlinna measures. The family includes examples with reducible denominators, and these yield counterexamples to the irreducibility assertion in a conjecture of Knese. We then characterize reducibility within this family both algebraically and geometrically.
ER  -