Stable polynomials and admissible numerators in product domains
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Soggetti: | |
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| _version_ | 1866913939363725312 |
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| author | Bickel, Kelly Knese, Greg Pascoe, James Eldred Sola, Alan |
| author_facet | Bickel, Kelly Knese, Greg Pascoe, James Eldred Sola, Alan |
| contents | Given a polynomial $p$ with no zeros in the polydisk, or equivalently the poly-upper half-plane, we study the problem of determining the ideal of polynomials $q$ with the property that the rational function $q/p$ is bounded near a boundary zero of $p$. We give a complete description of this ideal of numerators in the case where the zero set of $p$ is smooth and satisfies a non-degeneracy condition. We also give a description of the ideal in terms of an integral closure when $p$ has an isolated zero on the distinguished boundary. Constructions of multivariate stable polynomials are presented to illustrate sharpness of our results and necessity of our assumptions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_13014 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stable polynomials and admissible numerators in product domains Bickel, Kelly Knese, Greg Pascoe, James Eldred Sola, Alan Complex Variables Algebraic Geometry Classical Analysis and ODEs 13A15, 13B22, 32A40, 32B05 Given a polynomial $p$ with no zeros in the polydisk, or equivalently the poly-upper half-plane, we study the problem of determining the ideal of polynomials $q$ with the property that the rational function $q/p$ is bounded near a boundary zero of $p$. We give a complete description of this ideal of numerators in the case where the zero set of $p$ is smooth and satisfies a non-degeneracy condition. We also give a description of the ideal in terms of an integral closure when $p$ has an isolated zero on the distinguished boundary. Constructions of multivariate stable polynomials are presented to illustrate sharpness of our results and necessity of our assumptions. |
| title | Stable polynomials and admissible numerators in product domains |
| topic | Complex Variables Algebraic Geometry Classical Analysis and ODEs 13A15, 13B22, 32A40, 32B05 |
| url | https://arxiv.org/abs/2406.13014 |