Pairing of zeros and critical points for random meromorphic functions on riemann surfaces

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Abstract

We prove that zeros and critical points of a random polynomial pN of degree N in one complex variable appear in pairs. More precisely, suppose pN is conditioned to have pN(ξ)=0 for a fixed ξεC. For εε (0, 1/ 2) we prove that there is a unique critical point in the annulus {z ∈ C/N-1-ε</z -ξ/< N-1+ε}and no critical points closer to ξ with probability at least 1-O(N-3/2+3ε). We also prove an analogous statement in the more general setting of random meromorphic functions on a closed Riemann surface.

Original languageEnglish (US)
Pages (from-to)111-140
Number of pages30
JournalMathematical Research Letters
Volume22
Issue number1
DOIs
StatePublished - 2015
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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