Integral geometry and real zeros of Thue-Morse polynomials

We study the average number of intersecting points of a given curve with ra ndom hyperplanes in an n-dimensional Euclidean space. As noticed by A. Edel man and E. Kostlan, this problem is closely linked to finding the average n umber of real zeros of random polynomials. They showed that a real polynomi al of degree n has on average 2/pi log n + O(1) real zeros (M. Kac's theore m). This result leads us to the following problem: given a real sequence (alpha (k))(k is an element ofN), study the average 1/N Sigma (N-1)(n=0) rho (f(n)), where rho (f(n)) is the number of real zeros of f(n)(X) = alpha (0) + alpha X-1 + ... + alpha X-n(n). We give theoretical results for the Thue-Morse p olynomials and numerical evidence for other polynomials.