Random polynomials with a prescribed number of real zeros
Teoriâ veroâtnostej i ee primeneniâ, Tome 47 (2002) no. 3, pp. 600-606
Voir la notice de l'article provenant de la source Math-Net.Ru
Consider a polynomial of the form $a_0+a_1x+\dots+a_nx^n$ with random coefficients $a_j$. It is shown that, under mild conditions, one can choose the distributions of the aj from a given class of distributions so that, with probability arbitrarily close to 1, the random polynomial has a prescribed number of real roots. The proof is based on the gliding hump method. It is also shown how this method can be used to solve related problems for random sums of orthogonal polynomials and random power series.
Keywords:
number of real roots, random algebraic polynomials, random power series, gliding hump method.
@article{TVP_2002_47_3_a14,
author = {L. A. Imhof},
title = {Random polynomials with a prescribed number of real zeros},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {600--606},
publisher = {mathdoc},
volume = {47},
number = {3},
year = {2002},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TVP_2002_47_3_a14/}
}
L. A. Imhof. Random polynomials with a prescribed number of real zeros. Teoriâ veroâtnostej i ee primeneniâ, Tome 47 (2002) no. 3, pp. 600-606. http://geodesic.mathdoc.fr/item/TVP_2002_47_3_a14/