Concentration for the zero set of large random polynomial systems

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 $N\to\infty$....

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Bibliographic Details
Main Author Subag, Eliran
Format Journal Article
LanguageEnglish
Published 21.03.2023
Subjects
Mathematics - Probability
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Summary: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 $K$.
DOI:10.48550/arxiv.2303.11924