Integral geometry and real zeros of Thue-Morse polynomials

Abstract

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.