An Example of a Random Polynomial with Unusual Zeros Behavior

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MR2223217 (2007i:60061)60G99Zaporozhets, D. N.(RS-AOS2)An example of a random polynomial with unusual behavior of the roots.(Russian. Russian summary)Teor. Veroyatn. Primen.50 (2005),no. 3,549–555;translation in Theory Probab. Appl.50(2006),no. 3,529–535.

Let Mn be the number of real zeros of a polynomial

Gn(t) =n∑

k=0

ξk tk

with i.i.d. random coefficientsξk. For all previously studied examples, it is true thatEMn ≈ log n.It was even conjectured that this asymptotics should be true for any fixed nondegenerated law ofthe coefficients. However, the author here constructs a common law for the variablesξk such that

supn

EMn <∞.

Reviewed byMikhail A. Lifshits

c© Copyright American Mathematical Society 2007, 2012

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An Example of a Random Polynomial with Unusual Zeros Behavior

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