 
Summary: Complexity of Real Root Isolation Using Continued
Fractions
Vikram Sharma
GALAAD, INRIA
Sophia Antipolis, France
vikram.sharma@sophia.inria.fr
January 25, 2007
Abstract
Akritas had proposed an algorithm, which utilizes the continued fraction expansion
of real algebraic numbers, for isolating the real roots of a univariate polynomial. The
efficiency of the algorithm depends upon computing tight lower bounds on the smallest
positive root of a polynomial. The known complexity bounds for the algorithm rely
on the impractical assumption that it is possible to efficiently compute a lower bound
that is at a constant distance from the smallest positive root; without this assumption,
the worst case bounds are exponential. In this paper, we derive a polynomial worst
case bound on the algorithm without relying on the above mentioned assumption. In
particular, we show that for a squarefree integer polynomial of degree n and coefficients
of bitlength L, the bitcomplexity of the algorithm is O(n8L3), where O indicates that
we are omitting logarithmic factors.
1 Introduction
