Slices of Stable Polynomials and Connections to the Grace-Walsh-Szegő theorem
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| Format: | Preprint |
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2024
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| _version_ | 1866913976702468096 |
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| author | Debus, Sebastian Riener, Cordian Schabert, Robin |
| author_facet | Debus, Sebastian Riener, Cordian Schabert, Robin |
| contents | Univariate polynomials are called stable with respect to a domain $D$ if all of their roots lie in $D$. We study linear slices of the space of stable univariate polynomials with respect to a half-plane. We show that a linear slice always contains a stable polynomial with only a few distinct roots. Subsequently, we apply these results to symmetric polynomials and varieties. We show that for varieties defined by few multiaffine symmetric polynomials, the existence of a point in $D^n$ with few distinct coordinates is necessary and sufficient for the intersection with $D^n$ to be non-empty. This is at the same time a generalization of the so-called degree principle to stable polynomials and a result similar to Grace-Walsh-Szegő's coincidence theorem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_05905 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Slices of Stable Polynomials and Connections to the Grace-Walsh-Szegő theorem Debus, Sebastian Riener, Cordian Schabert, Robin Algebraic Geometry 12D10, 20C30, 26C10, 14P10 Univariate polynomials are called stable with respect to a domain $D$ if all of their roots lie in $D$. We study linear slices of the space of stable univariate polynomials with respect to a half-plane. We show that a linear slice always contains a stable polynomial with only a few distinct roots. Subsequently, we apply these results to symmetric polynomials and varieties. We show that for varieties defined by few multiaffine symmetric polynomials, the existence of a point in $D^n$ with few distinct coordinates is necessary and sufficient for the intersection with $D^n$ to be non-empty. This is at the same time a generalization of the so-called degree principle to stable polynomials and a result similar to Grace-Walsh-Szegő's coincidence theorem. |
| title | Slices of Stable Polynomials and Connections to the Grace-Walsh-Szegő theorem |
| topic | Algebraic Geometry 12D10, 20C30, 26C10, 14P10 |
| url | https://arxiv.org/abs/2402.05905 |