Almost tight recursion tree bounds for the Descartes method

Authors:
Arno Eigenwillig;Vikram Sharma;Chee K. Yap
Affiliations:
Max-Planck-Institut für Informatik, Saarbrücken, Germany;NYU, New York;NYU, New York
Venue:
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Year:
2006

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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) = Εni=0 aiXi with integer coefficients |ai| 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(logn), and we use it to derive the best known bit complexity bound for the integer case.