Integral geometry and real zeros of Thue-Morse polynomials

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 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 1/N N-1∑n=0 ρ(f_n), where ρ(f_n) is the number of real zeros of f_n(X) = α₀+α₁X+⋯+ α_nXⁿ. We give theoretical results for the Thue-Morse polynomials and numerical evidence for other polynomials.