| Autor: |
Do, Yen, Nguyen, Hoi, Vu, Van |
| Předmět: |
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| Zdroj: |
Proceedings of the London Mathematical Society; Dec2015, Vol. 111 Issue 6, p1231-1260, 30p |
| Abstrakt: |
Let Pn(x) = Σi=0n ξixi be a Kac random polynomial where the coefficients ξi are i.i.d. copies of a given random variable ξ. Our main result is an optimal quantitative bound concerning real roots repulsion. This leads to an optimal bound on the probability that there is a real double root. As an application, we consider the problem of estimating the number of real roots of Pn, which has a long history and in particular was the main subject of a celebrated series of papers by Littlewood and Offord from the 1940s. We show, for a large and natural family of atom variables ξ, that the expected number of real roots of Pn(x) is exactly (2/π) log n + C + o(1), where C is an absolute constant depending on the atom variable ξ. Prior to this paper, such a result was known only for the case when ξ is Gaussian. [ABSTRACT FROM AUTHOR] |
| Databáze: |
Complementary Index |
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