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Computational Aspects of Modular Forms and Galois RepresentationsHow One Can Compute in Polynomial Time the Value of Ramanujan's Tau at a Prime (AM-176)$
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Bas Edixhoven and Jean-Marc Couveignes

Print publication date: 2011

Print ISBN-13: 9780691142012

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691142012.001.0001

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Computing complex zeros of polynomials and power series

Computing complex zeros of polynomials and power series

Chapter:
(p.95) Chapter Five Computing complex zeros of polynomials and power series
Source:
Computational Aspects of Modular Forms and Galois Representations
Author(s):

Jean-Marc Couveignes

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691142012.003.0005

The purpose of this chapter is twofold. First, it will prove two theorems (5.3.1 and 5.4.2) about the complexity of computing complex roots of polynomials and zeros of power series. The existence of a deterministic polynomial time algorithm for these purposes plays an important role in this book. More important, it will also explain what it means to compute with real or complex data in polynomial time. The chapter first recalls basic definitions in computational complexity theory, it then deals with the problem of computing square roots. The more general problem of computing complex roots of polynomials is treated thereafter and, finally, the chapter studies the problem of finding zeros of a converging power series.

Keywords:   modular forms, polynomials, complex roots, polynomial time, power series, square root

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