## 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 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) = α_{0}+α_{1}X+⋯+ α_{n}X^{n}. We give theoretical results for the Thue-Morse polynomials and numerical evidence for other polynomials.

Original language | English |
---|---|

Pages (from-to) | 339-350 |

Number of pages | 12 |

Journal | Experimental Mathematics |

Volume | 9 |

Issue number | 3 |

Publication status | Published - 2000 |

Externally published | Yes |

## Keywords

- Integral geometry
- Real roots
- Thue-Morse sequence