Amortized Bound for Root Isolation via Sturm Sequences

  • Zilin Du
  • Vikram Sharma
  • Chee K. Yap
Part of the Trends in Mathematics book series (TM)

Abstract

This paper presents two results on the complexity of root isolation via Sturm sequences. Both results exploit amortization arguments.

For a square-free polynomial A (X) of degree d with L-bit integer coefficients, we use an amortization argument to show that all the roots, real or complex, can be isolated using at most O(dL + dlgd) Sturm probes. This extends Davenport’s result for the case of isolating all real roots.

We also show that a relatively straightforward algorithm, based on the classical subresultant PQS, allows us to evaluate the Sturm sequence of A(X) at rational Õ(dL)-bit values in time Õ(d3L); here the Õ-notation means we ignore logarithmic factors. Again, an amortization argument is used. We provide a family of examples to show that such amortization is necessary.

Keywords

Sturm sequence Davenport-Mahler bound subresultant complexity root isolation separation bound 

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2007

Authors and Affiliations

  • Zilin Du
    • 1
  • Vikram Sharma
    • 1
  • Chee K. Yap
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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