Concentration for the zero set of large random polynomial systems
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866929347284172800 |
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| author | Subag, Eliran |
| author_facet | Subag, Eliran |
| contents | For random systems of $K$ polynomials in $N + 1$ real variables which include the models of Kostlan (1987) and Shub and Smale (1993), we prove that the number of zeros on the unit sphere for $K = N$ or the Hausdorff measure of the zero set for $K < N$ concentrates around its mean as $N\to\infty$. To prove concentration we show that the variance of the latter random variable normalized by its mean goes to zero. The polynomial systems we consider depend on a set of parameters which determine the variance of their Gaussian coefficients. We prove that the convergence is uniform in those parameters and $K$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_11924 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Concentration for the zero set of large random polynomial systems Subag, Eliran Probability For random systems of $K$ polynomials in $N + 1$ real variables which include the models of Kostlan (1987) and Shub and Smale (1993), we prove that the number of zeros on the unit sphere for $K = N$ or the Hausdorff measure of the zero set for $K < N$ concentrates around its mean as $N\to\infty$. To prove concentration we show that the variance of the latter random variable normalized by its mean goes to zero. The polynomial systems we consider depend on a set of parameters which determine the variance of their Gaussian coefficients. We prove that the convergence is uniform in those parameters and $K$. |
| title | Concentration for the zero set of large random polynomial systems |
| topic | Probability |
| url | https://arxiv.org/abs/2303.11924 |