Random polynomials having few or no real zeros

By Amir Dembo, Bjorn Poonen, Qi-Man Shao, Ofer Zeitouni. 2000

84 citations recorded by the source corpus.

Consider a polynomial of large degree n whose coefficients are independent, identically distributed, nondegenerate random variables having zero mean and finite moments of all orders. We show that such a polynomial has exactly k real zeros with probability n−b+o(1) as n → ∞ through integers of the same parity as the fixed integer k ≥ 0. In particular, the probability that a random polynomial of large even degree n has no real zeros is n−b+o(1). The finite, positive constant b is characterized via the centered, stationary Gaussian process of correlation function sech(t/2). The value of b depends neither on k nor upon the specific law of the coefficients. Under an extra smoothness assumption about the law of the coefficients, with probability n−b+o(1) one may specify also the approximate locations of the k zeros on the real line. The constant b is replaced by b/2 in case the i.i.d. coefficients have a nonzero mean.

Subfields: Geometry and Topology, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/ecc29239883ffbbb99b4bc25d7737e4bc79e39c7

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Citations in the corpus

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No zero-crossings for random polynomials and the heat equation (2012, background): “Here bα = 0 when α ≤ −1 and otherwise bα ∈ (0,∞) is independent of the detailed regularly varying variance function and corresponds to persistence probabilities for an explicit stationary Gaussian process of smooth sample path.”
Local Universality of Zeroes of Random Polynomials (2013, background): “39, 40] also proved that if P(ξ= 0) = 0 and E|ξ|2+ε <∞ for some constant ε>0 then the variance of NR is (4 π(1 − 2 π) + o(1))logn, and furthermore established a central limit theorem for NR. In [10], Dembo et. al. computed the probability that NR = kfor any ﬁxed k. There are also some non-triv”