372 research outputs found
A complex analogue of Toda's Theorem
Toda \cite{Toda} proved in 1989 that the (discrete) polynomial time
hierarchy, , is contained in the class \mathbf{P}^{#\mathbf{P}},
namely the class of languages that can be decided by a Turing machine in
polynomial time given access to an oracle with the power to compute a function
in the counting complexity class #\mathbf{P}. This result, which illustrates
the power of counting is considered to be a seminal result in computational
complexity theory. An analogous result (with a compactness hypothesis) in the
complexity theory over the reals (in the sense of Blum-Shub-Smale real machines
\cite{BSS89}) was proved in \cite{BZ09}. Unlike Toda's proof in the discrete
case, which relied on sophisticated combinatorial arguments, the proof in
\cite{BZ09} is topological in nature in which the properties of the topological
join is used in a fundamental way. However, the constructions used in
\cite{BZ09} were semi-algebraic -- they used real inequalities in an essential
way and as such do not extend to the complex case. In this paper, we extend the
techniques developed in \cite{BZ09} to the complex projective case. A key role
is played by the complex join of quasi-projective complex varieties. As a
consequence we obtain a complex analogue of Toda's theorem. The results
contained in this paper, taken together with those contained in \cite{BZ09},
illustrate the central role of the Poincar\'e polynomial in algorithmic
algebraic geometry, as well as, in computational complexity theory over the
complex and real numbers -- namely, the ability to compute it efficiently
enables one to decide in polynomial time all languages in the (compact)
polynomial hierarchy over the appropriate field.Comment: 31 pages. Final version to appear in Foundations of Computational
Mathematic
Some Speed-Ups and Speed Limits for Real Algebraic Geometry
We give new positive and negative results (some conditional) on speeding up
computational algebraic geometry over the reals: (1) A new and sharper upper
bound on the number of connected components of a semialgebraic set. Our bound
is novel in that it is stated in terms of the volumes of certain polytopes and,
for a large class of inputs, beats the best previous bounds by a factor
exponential in the number of variables. (2) A new algorithm for approximating
the real roots of certain sparse polynomial systems. Two features of our
algorithm are (a) arithmetic complexity polylogarithmic in the degree of the
underlying complex variety (as opposed to the super-linear dependence in
earlier algorithms) and (b) a simple and efficient generalization to certain
univariate exponential sums. (3) Detecting whether a real algebraic surface
(given as the common zero set of some input straight-line programs) is not
smooth can be done in polynomial time within the classical Turing model (resp.
BSS model over C) only if P=NP (resp. NP<=BPP). The last result follows easily
from an unpublished result of Steve Smale.Comment: This is the final journal version which will appear in Journal of
Complexity. More typos are corrected, and a new section is added where the
bounds here are compared to an earlier result of Benedetti, Loeser, and
Risler. The LaTeX source needs the ajour.cls macro file to compil
Multiplicity Estimates: a Morse-theoretic approach
The problem of estimating the multiplicity of the zero of a polynomial when
restricted to the trajectory of a non-singular polynomial vector field, at one
or several points, has been considered by authors in several different fields.
The two best (incomparable) estimates are due to Gabrielov and Nesterenko.
In this paper we present a refinement of Gabrielov's method which
simultaneously improves these two estimates. Moreover, we give a geometric
description of the multiplicity function in terms certain naturally associated
polar varieties, giving a topological explanation for an asymptotic phenomenon
that was previously obtained by elimination theoretic methods in the works of
Brownawell, Masser and Nesterenko. We also give estimates in terms of Newton
polytopes, strongly generalizing the classical estimates.Comment: Minor revision; To appear in Duke Math. Journa
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