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PreprintarXiv licencearXiv

Higher-dimensional singularities of rational inner functions

  • Leonora Krajina
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Abstract

We study singularities of rational inner functions (RIFs) ฯ• in three and more variables. We focus on curve and higher-dimensional singularities of ฯ• in ๐•‹n and we show that, under certain conditions, derivative integrability conditions of partial derivatives of ฯ•=q/p stay constant along irreducible components of the zero set of p, and that the presence of vertical line singularities does not affect global integrability. This answers Question 2 in arXiv:1906.10913. Those results are then generalized to n variables. Finally, we consider slice matrices, which we use to obtain derivative integrability conditions of unidirectional compositions ฯ•N. We present several new examples, including an RIF with two vertical line singularities and one with vertical surface singularities.

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Cite

BibTeXRIS
BibTeX
@misc{krajina2026higherdimensional,
  title = {{Higher-dimensional singularities of rational inner functions}},
  author = {Leonora Krajina},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26805},
  eprint = {2609.26805},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26805},
}
RIS
TY  - UNPB
TI  - Higher-dimensional singularities of rational inner functions
AU  - Leonora Krajina
PY  - 2026
DA  - 2026/09/13
PB  - arXiv
DO  - 10.48550/arxiv.2609.26805
UR  - https://arxiv.org/abs/2609.26805
AB  - We study singularities of rational inner functions (RIFs) ฯ• in three and more variables. We focus on curve and higher-dimensional singularities of ฯ• in ๐•‹^n and we show that, under certain conditions, derivative integrability conditions of partial derivatives of ฯ•=q/p stay constant along irreducible components of the zero set of p, and that the presence of vertical line singularities does not affect global integrability. This answers Question 2 in arXiv:1906.10913. Those results are then generalized to n variables. Finally, we consider slice matrices, which we use to obtain derivative integrability conditions of unidirectional compositions ฯ•^N. We present several new examples, including an RIF with two vertical line singularities and one with vertical surface singularities.
ER  -