145 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
Complex Multiplication Tests for Elliptic Curves
We consider the problem of checking whether an elliptic curve defined over a
given number field has complex multiplication. We study two polynomial time
algorithms for this problem, one randomized and the other deterministic. The
randomized algorithm can be adapted to yield the discriminant of the
endomorphism ring of the curve.Comment: 13 pages, 2 tables, 1 appendi
Quantitative behavior of unipotent flows and an effective avoidance principle
We give an effective bound on how much time orbits of a unipotent group
on an arithmetic quotient can stay near homogeneous subvarieties of
corresponding to -subgroups of . In particular, we
show that if such a -orbit is moderately near a proper homogeneous
subvariety of for a long time it is very near a different
homogeneous subvariety. Our work builds upon the linearization method of Dani
and Margulis.
Our motivation in developing these bounds is in order to prove quantitative
density statements about unipotent orbits, which we plan to pursue in a
subsequent paper. New qualitative implications of our effective bounds are also
given.Comment: 52 page
On the Complexity of the Orbit Problem
We consider higher-dimensional versions of Kannan and Lipton's Orbit
Problem---determining whether a target vector space V may be reached from a
starting point x under repeated applications of a linear transformation A.
Answering two questions posed by Kannan and Lipton in the 1980s, we show that
when V has dimension one, this problem is solvable in polynomial time, and when
V has dimension two or three, the problem is in NP^{RP}
Calculating the power residue symbol and ibeta: Applications of computing the group structure of the principal units of a p-adic number field completion
In the recent PhD thesis of Bouw, an algorithm is examined that computes the group structure of the principal units of a p-adic number field completion. In the same thesis, this algorithm is used to compute Hilbert norm residue symbols. In the present paper, we will demonstrate two other applications. The first application is the computation of an important invariant of number field completions, called ibeta. The algorithm that computes ibeta is deterministic and runs in polynomial time. The second application
Computing zeta functions of large polynomial systems over finite fields
In this paper, we improve the algorithms of Lauder-Wan \cite{LW} and Harvey
\cite{Ha} to compute the zeta function of a system of polynomial equations
in variables over the finite field \FF_q of elements, for large.
The dependence on in the original algorithms was exponential in . Our
main result is a reduction of the exponential dependence on to a polynomial
dependence on . As an application, we speed up a doubly exponential time
algorithm from a software verification paper \cite{BJK} (on universal
equivalence of programs over finite fields) to singly exponential time. One key
new ingredient is an effective version of the classical Kronecker theorem which
(set-theoretically) reduces the number of defining equations for a "large"
polynomial system over \FF_q when is suitably large
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