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キーワード:
Computer Science, Symbolic Computation, cs.SC
要旨:
We present an algorithm for isolating the roots of an arbitrary complex
polynomial $p$ that also works for polynomials with multiple roots provided
that the number $k$ of distinct roots is given as part of the input. It outputs
$k$ pairwise disjoint disks each containing one of the distinct roots of $p$,
and its multiplicity. The algorithm uses approximate factorization as a
subroutine.
In addition, we apply the new root isolation algorithm to a recent algorithm
for computing the topology of a real planar algebraic curve specified as the
zero set of a bivariate integer polynomial and for isolating the real solutions
of a bivariate polynomial system. For input polynomials of degree $n$ and
bitsize $\tau$, we improve the currently best running time from
$\tO(n^{9}\tau+n^{8}\tau^{2})$ (deterministic) to $\tO(n^{6}+n^{5}\tau)$
(randomized) for topology computation and from $\tO(n^{8}+n^{7}\tau)$
(deterministic) to $\tO(n^{6}+n^{5}\tau)$ (randomized) for solving bivariate
systems.
