On the Boolean complexity of real root refinement

Victor Pan 1 Elias Tsigaridas 2
2 PolSys - Polynomial Systems
LIP6 - Laboratoire d'Informatique de Paris 6, Inria Paris-Rocquencourt
Abstract : We assume that a real square-free polynomial $A$ has a degree $d$, a maximum coefficient bitsize $\tau$ and a real root lying in an isolating interval and having no nonreal roots nearby (we quantify this assumption). Then, we combine the {\em Double Exponential Sieve} algorithm (also called the {\em Bisection of the Exponents}), the bisection, and Newton iteration to decrease the width of this inclusion interval by a factor of $t=2^{-L}$. The algorithm has Boolean complexity ${\widetilde{\mathcal{O}}_B}(d^2 \tau + d L )$. Our algorithms support the same complexity bound for the refinement of $r$ roots, for any $r\le d$.
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Victor Pan, Elias Tsigaridas. On the Boolean complexity of real root refinement. ISSAC 2013 - International Symposium on Symbolic and Algebraic Computation, Jun 2013, Boston, United States. ⟨10.1145/2465506.2465938⟩. ⟨hal-00816214v3⟩

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