### 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 |

Pages | 71-78 |

Number of pages | 8 |

Volume | 2006 |

State | Published - 2006 |

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

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Computer Science(all)

### Cite this

*Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation, ISSAC 2006*(Vol. 2006, pp. 71-78)

**Almost tight recursion tree bounds for the descartes method.** / Eigenwillig, Arno; Sharma, Vikram; Yap, Chee K.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation, ISSAC 2006.*vol. 2006, pp. 71-78, International Symposium on Symbolic and Algebraic Computation, ISSAC 2006, Genova, Italy, 7/9/06.

}

TY - GEN

T1 - Almost tight recursion tree bounds for the descartes method

AU - Eigenwillig, Arno

AU - Sharma, Vikram

AU - Yap, Chee K.

PY - 2006

Y1 - 2006

N2 - 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 aiXi with integer coefficients |a i| < 2L, 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.

AB - 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 aiXi with integer coefficients |a i| < 2L, 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.

KW - Bernstein basis

KW - Davenport-Mahler bound

KW - Descartes method

KW - Descartes rule of signs

KW - Polynomial real root isolation

UR - http://www.scopus.com/inward/record.url?scp=33748959696&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33748959696&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:33748959696

SN - 1595932763

SN - 9781595932761

VL - 2006

SP - 71

EP - 78

BT - Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation, ISSAC 2006

ER -