### Abstract

In this article we investigate real roots of real polynomials. By results of M. Kac [8-10] we know that a polynomial of degree n has on average (2/π)log n real zeros. See also results of Edelman and Kostlan [6] on the same subject. Some 10 years later Erdös and Offord [7] proved that the mean number of real roots of a random polynomial of degree n with coefficients ±1 is again (2/π) log n. This leads us to the following question: can we find sequences (α_{i})_{i∈ℕ} with coefficients ±1 such that the corresponding polynomials Σ^{n}_{i=0}α_{i}X^{i} have O(logn) real roots, and are these sequences random in some sense? We introduce generalized Thue-Morse sequences whose corresponding polynomials of large degree n have at least C log n real roots, where C is an explicit positive constant. Finally, we discuss the spectral measure of these sequences.

Original language | English |
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Pages (from-to) | 309-319 |

Number of pages | 11 |

Journal | Acta Arithmetica |

Volume | 99 |

Issue number | 4 |

Publication status | Published - 2001 |

Externally published | Yes |

### Keywords

- Real roots of polynomials
- Spectral measure
- Thue-Morse sequence

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## Cite this

*Acta Arithmetica*,

*99*(4), 309-319.