We study the average number of intersecting points of a given curve with random hyperplanes in an $n$-dimensional Euclidean space. As noticed by A. Edelman and E. Kostlan, this problem is closely linked to finding the average number of real zeros of random polynomials. They showed that a real polynomial of degree $n$ has on average $\frac{2}{\pi}\log n +O(1)$ real zeros (M. Kac's theorem).
¶ This result leads us to the following problem: given a real sequence $(\alpha_k)_{k\in\N }$, study the average $$\frac{1}{N}\sum_{n=0}^{N-1} \rho(f_{n}),$$ where $\rho(f_n)$ is the number of real zeros of $f_n(X)=\alpha_0+\alpha_1X+\cdots+\alpha_nX^n$. We give theoretical results for the Thue-Morse polynomials and numerical evidence for other polynomials.
Publié le : 2000-05-14
Classification:
Integral geometry,
Thue-Morse sequence,
real roots,
11K99,
11K45,
26C10
@article{1045604669,
author = {Doche, Christophe and France, Michel Mend\`es},
title = {Integral geometry and real zeros of Thue-Morse polynomials},
journal = {Experiment. Math.},
volume = {9},
number = {3},
year = {2000},
pages = { 339-350},
language = {en},
url = {http://dml.mathdoc.fr/item/1045604669}
}
Doche, Christophe; France, Michel Mendès. Integral geometry and real zeros of Thue-Morse polynomials. Experiment. Math., Tome 9 (2000) no. 3, pp. 339-350. http://gdmltest.u-ga.fr/item/1045604669/