Integral geometry and real zeros of Thue-Morse polynomials

Christophe Doche*, Michel Mendès France

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


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 (αk)k∈ℕ, study the average 1/N N-1∑n=0 ρ(fn), where ρ(fn) is the number of real zeros of fn(X) = α01X+⋯+ αnXn. We give theoretical results for the Thue-Morse polynomials and numerical evidence for other polynomials.

Original languageEnglish
Pages (from-to)339-350
Number of pages12
JournalExperimental Mathematics
Issue number3
Publication statusPublished - 2000
Externally publishedYes


  • Integral geometry
  • Real roots
  • Thue-Morse sequence


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