## Abstract

We consider the zero sets Z_{N} of systems of m random polynomials of degree N in m complex variables, and we give asymptotic formulas for the random variables given by summing a smooth test function over Z_{N}. Our asymptotic formulas show that the variances for these smooth statistics have the growth N^{m-2}. We also prove analogues for the integrals of smooth test forms over the subvarieties dened by k < m random polynomials. Such linear statistics of random zero sets are smooth analogues of the random variables given by counting the number of zeros in an open set, which we proved elsewhere to have variances of order N^{m-1/2}. We use the variance a symptotics and off-diagonal estimates of Szego{double acute} kernels to extend the central limit theorem of Sodin-Tsirelson to the case of smooth linear statistics for zero sets of codimension one in any dimension m.

Original language | English (US) |
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Pages (from-to) | 1145-1167 |

Number of pages | 23 |

Journal | Pure and Applied Mathematics Quarterly |

Volume | 6 |

Issue number | 4 |

DOIs | |

State | Published - Oct 2010 |

## Keywords

- Käahler manifold
- Positive line bundle
- Random holomorphic sections
- Szego{double acute} kernel
- Zeros of random polynomials

## ASJC Scopus subject areas

- Mathematics(all)