ZEROS OF RANDOM POLYNOMIALS ON C m
ZEROS OF RANDOM POLYNOMIALS ON C m
复制标题
C m 上的随机多项式的零点
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
B. Shiffman
中科院分区：
文献类型：
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作者：
T. Bloom;B. Shiffman
对于常规紧凑型套装$ k $ in $ mathbb {c}^m $和$ k $上满足Bernstein-Markov不平等的量子$ MU $，我们认为合奏$ MATHCAL $ MATHCAL {p} _n $ n $，赋予$ l^2（MU）$引起的高斯概率度量。我们表明，对于$ n $，$ m $ polyenmials in $ natcal {p} _n $倾向于集中在$ k $的silov边界周围；更确切地说，它们的预期分布是渐近的，是$ n^m mu_ {eq} $，其中$ mu_ {eq} $是$ k $的平衡度量。对于$ k $是单位球的情况，我们将预期分布的缩放级别渐近性为$ n oinfty $。
For a regular compact set $K$ in $mathbb{C}^m$ and a measure $mu$ on $K$ satisfying the Bernstein-Markov inequality, we consider the ensemble $mathcal{P}_N$ of polynomials of degree $N$, endowed with the Gaussian probability measure induced by $L^2(mu)$. We show that for large $N$, the simultaneous zeros of $m$ polynomials in $mathcal{P}_N$ tend to concentrate around the Silov boundary of $K$; more precisely, their expected distribution is asymptotic to $N^m mu_{eq}$, where $mu_{eq}$ is the equilibrium measure of $K$. For the case where $K$ is the unit ball, we give scaling asymptotics for the expected distribution of zeros as $N oinfty$.end