Abstract
Let pN be a random degree N polynomial in one complex variable whose zeros are chosen independently from a fixed probability measure μ on the Riemann sphere S2. This article proves that if we condition pN to have a zero at some fixed point ζ ϵ S2, then, with high probability, there will be a critical point ωζ at a distance N-1 away from ζ. This N-1 distance is much smaller than the N-1/2 typical spacing between nearest neighbors for N i.i.d. points on S2. Moreover, with the same high probability, the argument of ωζ relative to ζ is a deterministic function of μ plus fluctuations on the order of N-1.
Original language | English (US) |
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Pages (from-to) | 1498-1511 |
Number of pages | 14 |
Journal | Annales de l'institut Henri Poincare (B) Probability and Statistics |
Volume | 53 |
Issue number | 3 |
DOIs | |
State | Published - Aug 2017 |
Externally published | Yes |
All Science Journal Classification (ASJC) codes
- Statistics and Probability
- Statistics, Probability and Uncertainty
Keywords
- Critical points
- Gauss-Lucas
- Random polynomials
- Zeros