49,518 research outputs found
Volume computation for polytopes and partition functions for classical root systems
This paper presents an algorithm to compute the value of the inverse Laplace
transforms of rational functions with poles on arrangements of hyperplanes. As
an application, we present an efficient computation of the partition function
for classical root systems.Comment: 55 pages, 14 figures. Maple programs available at
http://www.math.polytechnique.fr/~vergne/work/IntegralPoints.htm
High order symplectic integrators for perturbed Hamiltonian systems
We present a class of symplectic integrators adapted for the integration of
perturbed Hamiltonian systems of the form . We give a
constructive proof that for all integer , there exists an integrator with
positive steps with a remainder of order ,
where is the stepsize of the integrator. The analytical expressions of
the leading terms of the remainders are given at all orders. In many cases, a
corrector step can be performed such that the remainder becomes
. The performances of these integrators
are compared for the simple pendulum and the planetary 3-Body problem of
Sun-Jupiter-Saturn.Comment: 24 pages, 6 figurre
Discovering the roots: Uniform closure results for algebraic classes under factoring
Newton iteration (NI) is an almost 350 years old recursive formula that
approximates a simple root of a polynomial quite rapidly. We generalize it to a
matrix recurrence (allRootsNI) that approximates all the roots simultaneously.
In this form, the process yields a better circuit complexity in the case when
the number of roots is small but the multiplicities are exponentially
large. Our method sets up a linear system in unknowns and iteratively
builds the roots as formal power series. For an algebraic circuit
of size we prove that each factor has size at most a
polynomial in: and the degree of the squarefree part of . Consequently,
if is a -hard polynomial then any nonzero multiple
is equally hard for arbitrary positive 's, assuming
that is at most .
It is an old open question whether the class of poly()-sized formulas
(resp. algebraic branching programs) is closed under factoring. We show that
given a polynomial of degree and formula (resp. ABP) size
we can find a similar size formula (resp. ABP) factor in
randomized poly()-time. Consequently, if determinant requires
size formula, then the same can be said about any of its
nonzero multiples.
As part of our proofs, we identify a new property of multivariate polynomial
factorization. We show that under a random linear transformation ,
completely factors via power series roots. Moreover, the
factorization adapts well to circuit complexity analysis. This with allRootsNI
are the techniques that help us make progress towards the old open problems,
supplementing the large body of classical results and concepts in algebraic
circuit factorization (eg. Zassenhaus, J.NT 1969, Kaltofen, STOC 1985-7 \&
Burgisser, FOCS 2001).Comment: 33 Pages, No figure
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