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- W905632657 abstract "AbstractIn this paper we find the asymptotic main term of the variance ofthe number of roots of Kostlan-Shub-Smale random polynomials and prove a central limit theorem for the number of roots as thedegree goes to infinity.Resum´eDans ce papier nous trouvons le terme asymptotique dominant de la variance du nombre de racinesr´eels des polynomes al´eatoires de Kostlan-Shub-Smale et d´emontrons un th´eor`eme de la limitecentrale pour ce nombre de racines. Consider the Kostlan-Shub-Smale (KSS for short) ensemble of random polynomials:X d (x) :=X dn=0 a (d)n x n ; x ∈ R,where d is the degree of the polynomial and the coefficients (a (d)n ) are independent centered Gaussianrandom variables whose variances are the binomial coefficients, more precisely Var(a (d)n ) = dn .Denote by N d the number of real roots of X d , that isN d := #{x ∈ R: X d (x) = 0}.It is well known that E(N d ) =√d [10, 15]. The aim of this paper is to prove the following result.Theorem 1. The variance of the number of real roots N" @default.
- W905632657 created "2016-06-24" @default.
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- W905632657 date "2015-04-21" @default.
- W905632657 modified "2023-09-27" @default.
- W905632657 title "CLT for the zeros of Kostlan Shub Smale random polynomials" @default.
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