Integral geometry and real zeros of Thue-Morse polynomials. (English) Zbl 0976.52005

Summary: 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 \((\alpha_k)_{k\in\mathbb{N}}\), study the average \[ \tfrac 1N\sum^{N-1}_{n=0}\rho(f_n), \] 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.


52A22 Random convex sets and integral geometry (aspects of convex geometry)
53C65 Integral geometry
11B85 Automata sequences
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