INTEGRAL GEOMETRY AND REAL ZEROS OF THUE–MORSE POLYNOMIALS

CHRISTOPHE DOCHE AND MICHEL MENDÈS FRANCE

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}{π} log n + O(1) real zeros (M. Kac’s theorem).

This result leads us to the following problem: given a real sequence (α_{k})_{k∈ℕ}, study the average

\frac{1}{N} ∑_{n=0}^{N−1} ρ(f_{n}),

where ρ(f_{n}) is the number of real zeros of f_{n}(X) = α_{0}+α_{1}X +⋯+ α_{n}X^{n}. We give theoretical results for the Thue–Morse polynomials and numerical evidence for other polynomials.