English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We consider the zero sets $Z_N$ of systems of $m$ random polynomials ofdegree $N$ in $m$ complex variables, and we give asymptotic formulas for therandom variables given by summing a smooth test function over $Z_N$. Ourasymptotic formulas show that the variances for these smooth statistics havethe growth $N^{m-2}$. We also prove analogues for the integrals of smooth testforms over the subvarieties defined by $k<m$ random polynomials. Such linearstatistics of random zero sets are smooth analogues of the random variablesgiven by counting the number of zeros in an open set, which we proved elsewhereto have variances of order $N^{m-1/2}$. We use the variance asymptotics andoff-diagonal estimates of Szego kernels to extend an asymptotic normalityresult of Sodin-Tsirelson to the case of smooth linear statistics for zero setsof codimension one in any dimension $m$.

Number variance of random zeros on complex manifolds, II: smooth statistics