Abstract
We show that the variance of the number of simultaneous zeros of m i.i.d. Gaussian random polynomials of degree N in an open set U ⊆Cm with smooth boundary is asymptotic to Nm-1/2νmm Vol ∂U , where νmm is a universal constant depending only on the dimension m. We also give formulas for the variance of the volume of the set of simultaneous zeros in U of k < m random degree-N polynomials on Cm . Our results hold more generally for the simultaneous zeros of random holomorphic sections of the N-th power of any positive line bundle over any m-dimensional compact Kähler manifold.
Original language | English (US) |
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Pages (from-to) | 1422-1475 |
Number of pages | 54 |
Journal | Geometric and Functional Analysis |
Volume | 18 |
Issue number | 4 |
DOIs | |
State | Published - Dec 2008 |
Keywords
- Holomorphic line bundle
- Kähler manifold
- Random holomorphic sections
- Szego″ kernel
- Zeros of random polynomials
ASJC Scopus subject areas
- Analysis
- Geometry and Topology