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Preprint
Local theory of stable polynomials and bounded rational functions of several variables
arXiv.org
15 Sep 2021
Abstract
We provide detailed local descriptions of stable polynomials in terms of
their homogeneous decompositions, Puiseux expansions, and transfer function
realizations. We use this theory to first prove that bounded rational functions
on the polydisk possess non-tangential limits at every boundary point. We
relate higher non-tangential regularity and distinguished boundary behavior of
bounded rational functions to geometric properties of the zero sets of stable
polynomials via our local descriptions. For a fixed stable polynomial $p$, we
analyze the ideal of numerators $q$ such that $q/p$ is bounded on the bi-upper
half plane. We completely characterize this ideal in several geometrically
interesting situations including smooth points, double points, and ordinary
multiple points of $p$. Finally, we analyze integrability properties of bounded
rational functions and their derivatives on the bidisk.
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Details
- Title
- Local theory of stable polynomials and bounded rational functions of several variables
- Creators
- Kelly BickelGreg KneseJames Eldred PascoeAlan Sola
- Publication Details
- arXiv.org
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991021880186804721