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 show that a real polynomial of degree n has in average log n + O(1) real zeros (M. Kac's theorem). This result leads us to the following question: given a real sequence (ff k ) k2N , to study the average P N \Gamma1 n=0 ae(f n ); where ae(f n ) is the number of real zeros of fn (X) = ff 0 + ff 1 X + \Delta \Delta \Delta + ff nX . Theoretical results are given for the Thue-Morse polynomials as well as numerical evidence for other polynomials