On the number of connected components of random algebraic hypersurfaces
On the number of connected components of random algebraic hypersurfaces
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DOI：
10.1016/j.geomphys.2015.04.006
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发表时间：
2014-04
影响因子：
1.5
通讯作者：
Y. Fyodorov;A. Lerário;Erik Lundberg
中科院分区：
文献类型：
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作者：
Y. Fyodorov;A. Lerário;Erik Lundberg
我们研究随机代数超曲面 X 的分量数 b 0 (X) 的期望，该随机代数超曲面 X 由 d 次随机齐次多项式 f 的射影空间 R P n 中的零集定义。具体来说，我们考虑不变系综，即在变量正交变化下不变的多项式高斯系综。固定n，在系综族的一些重新标度假设下（如d→∞），我们证明E b 0 (X) 与[E b 0 (X∩ R P 1)] n 具有相同的增长顺序。这将 X 的平均分量数与 M. Kac (1943) 关于随机单变量多项式 f| 的零点数量的经典问题联系起来。 R P 1. 证明需要 E b 0 (X) 的上限，我们通过使用 Fyodorov (2013) 的随机矩阵理论方法计算极值来获得，并且还需要下限，我们通过修改 Lerario 和 Lundberg (2015) 以及 Nazarov 和 Sodin (2009) 的屏障方法获得下限。我们还提供隐含常数的定量上限；对于真实的 Fubini–Study 模型，这些估计提供了 E b 0 (X) 的主导系数（以 d 为单位）的超指数衰减（如 n→∞）。
We study the expectation of the number of components b 0 (X) of a random algebraic hypersurface X defined by the zero set in projective space R P n of a random homogeneous polynomial f of degree d. Specifically, we consider invariant ensembles, that is Gaussian ensembles of polynomials that are invariant under an orthogonal change of variables. Fixing n, under some rescaling assumptions on the family of ensembles (as d→∞), we prove that E b 0 (X) has the same order of growth as [E b 0 (X∩ R P 1)] n. This relates the average number of components of X to the classical problem of M. Kac (1943) on the number of zeros of the random univariate polynomial f| R P 1. The proof requires an upper bound for E b 0 (X), which we obtain by counting extrema using Random Matrix Theory methods from Fyodorov (2013), and it also requires a lower bound, which we obtain by a modification of the barrier method from Lerario and Lundberg (2015) and Nazarov and Sodin (2009). We also provide quantitative upper bounds on implied constants; for the real Fubini–Study model these estimates provide super-exponential decay (as n→∞) of the leading coefficient (in d) of E b 0 (X).