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Journal of Symbolic Computation 41 (2006) 4966 www.elsevier.com/locate/jsc

Summary: Journal of Symbolic Computation 41 (2006) 49­66
New bounds for the Descartes method
Werner Krandicka,, Kurt Mehlhornb
a Department of Computer Science, Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104, USA
b Max-Planck-Institut für Informatik, Stuhlsatzenhausweg 85, 66123 Saarbrücken, Germany
Received 29 September 2004; accepted 8 February 2005
Available online 14 November 2005
We give a new bound for the number of recursive subdivisions in the Descartes method for polynomial
real root isolation. Our proof uses Ostrowski's theory of normal power series from 1950 which has so far
been overlooked in the literature. We combine Ostrowski's results with a theorem of Davenport from 1985
to obtain our bound. We also characterize normality of cubic polynomials by explicit conditions on their
roots and derive a generalization of one of Ostrowski's theorems.
c 2005 Elsevier Ltd. All rights reserved.
Keywords: Polynomial real root isolation; Descartes rule of signs; Modified Uspensky method; Recursion tree analysis;
Normal polynomials; Root separation bounds; History of mathematics; Möbius transformations; Coefficient sign
variations; Cylindrical algebraic decomposition
1. Introduction
Polynomial real root isolation is the task of computing disjoint intervals, each containing a


Source: Akritas, Alkiviadis G. - Department of Computer and Communication Engineering, University of Thessaly


Collections: Computer Technologies and Information Sciences