Characterization of the boundary zero set of stable polynomials in $\mathbb B_2$

Given a polynomial $p\in \mathbb{C}[z,w]$ whose zeros lie outside the unit ball $\mathbb{B}_2$ of $\mathbb{C}^2,$ we study its local Weierstrass polynomial--obtained via the Weierstrass Preparation Theorem--and provide a strong version of the Nash curve selection lemma. These results lead to a characterization of the intersection of the zero set of $p$ with the unit sphere: $\mathcal{Z}(p)\cap\mathbb{S}_{2}.$

Authors

Publication Details

Journal
arXiv (Cornell University)
Published
2026-11-02
DOI
https://doi.org/10.48550/arxiv.2611.02838
Primary Topic
Holomorphic and Operator Theory
Type
preprint
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preprint

Characterization of the boundary zero set of stable polynomials in $\mathbb B_2$

Dimitrios Vavitsas
arXiv (Cornell University)
Holomorphic and Operator Theory
preprint

Characterization of the boundary zero set of stable polynomials in $\mathbb B_2$

Dimitrios Vavitsas
preprint en

Abstract

Given a polynomial $p\in \mathbb{C}[z,w]$ whose zeros lie outside the unit ball $\mathbb{B}_2$ of $\mathbb{C}^2,$ we study its local Weierstrass polynomial--obtained via the Weierstrass Preparation Theorem--and provide a strong version of the Nash curve selection lemma. These results lead to a characterization of the intersection of the zero set of $p$ with the unit sphere: $\mathcal{Z}(p)\cap\mathbb{S}_{2}.$

arXiv (Cornell University)
Holomorphic and Operator Theory
DOIOpen Access PDFOpenAlex
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Characterization of the boundary zero set of stable polynomials in $\mathbb B_2$ — Dimitrios Vavitsas · arXiv (Cornell University) (2026) | TGRS Research Map | TGRS