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2000 Integral geometry and real zeros of Thue-Morse polynomials
Christophe Doche, Michel Mendès France
Experiment. Math. 9(3): 339-350 (2000).


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.


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Christophe Doche. Michel Mendès France. "Integral geometry and real zeros of Thue-Morse polynomials." Experiment. Math. 9 (3) 339 - 350, 2000.


Published: 2000
First available in Project Euclid: 18 February 2003

zbMATH: 0976.52005
MathSciNet: MR1795306

Primary: 11K99
Secondary: 11K45 , 26C10

Keywords: integral geometry , real roots , Thue-Morse sequence

Rights: Copyright © 2000 A K Peters, Ltd.


Vol.9 • No. 3 • 2000
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