### Abstract

We give a unified ("basis free") framework for the Descartes method for real root isolation of square-free real polynomials. This framework encompasses the usual Descartes' rule of sign method for polynomials in the power basis as well as its analog in the Bernstein basis. We then give a new bound on the size of the recursion tree in the Descartes method for polynomials with real coefficients. Applied to polynomials A(X) = ∑_{i=0} ^{n} a_{i}X^{i} with integer coefficients |a _{i}| < 2^{L}, this yields a bound of O(n(L + logn)) on the size of recursion trees. We show that this bound is tight for L = Ω(log n), and we use it to derive the best known bit complexity bound for the integer case.

Original language | English (US) |
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Title of host publication | Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation, ISSAC 2006 |

Publisher | Association for Computing Machinery (ACM) |

Pages | 71-78 |

Number of pages | 8 |

ISBN (Print) | 1595932763, 9781595932761 |

DOIs | |

State | Published - 2006 |

Event | International Symposium on Symbolic and Algebraic Computation, ISSAC 2006 - Genova, Italy Duration: Jul 9 2006 → Jul 12 2006 |

### Publication series

Name | Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC |
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Volume | 2006 |

### Other

Other | International Symposium on Symbolic and Algebraic Computation, ISSAC 2006 |
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Country | Italy |

City | Genova |

Period | 7/9/06 → 7/12/06 |

### Keywords

- Bernstein basis
- Davenport-Mahler bound
- Descartes method
- Descartes rule of signs
- Polynomial real root isolation

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation, ISSAC 2006*(pp. 71-78). (Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC; Vol. 2006). Association for Computing Machinery (ACM). https://doi.org/10.1145/1145768.1145786