64,039 research outputs found
A novel approach to integration by parts reduction
Integration by parts reduction is a standard component of most modern
multi-loop calculations in quantum field theory. We present a novel strategy
constructed to overcome the limitations of currently available reduction
programs based on Laporta's algorithm. The key idea is to construct algebraic
identities from numerical samples obtained from reductions over finite fields.
We expect the method to be highly amenable to parallelization, show a low
memory footprint during the reduction step, and allow for significantly better
run-times.Comment: 4 pages. Version 2 is the final, published version of this articl
Software for Exact Integration of Polynomials over Polyhedra
We are interested in the fast computation of the exact value of integrals of
polynomial functions over convex polyhedra. We present speed ups and extensions
of the algorithms presented in previous work. We present the new software
implementation and provide benchmark computations. The computation of integrals
of polynomials over polyhedral regions has many applications; here we
demonstrate our algorithmic tools solving a challenge from combinatorial voting
theory.Comment: Major updat
Reconstructing Rational Functions with
We present the open-source library for the
reconstruction of multivariate rational functions over finite fields. We
discuss the involved algorithms and their implementation. As an application, we
use in the context of integration-by-parts reductions and
compare runtime and memory consumption to a fully algebraic approach with the
program .Comment: 46 pages, 3 figures, 6 tables; v2: matches published versio
Approximate computations with modular curves
This article gives an introduction for mathematicians interested in numerical
computations in algebraic geometry and number theory to some recent progress in
algorithmic number theory, emphasising the key role of approximate computations
with modular curves and their Jacobians. These approximations are done in
polynomial time in the dimension and the required number of significant digits.
We explain the main ideas of how the approximations are done, illustrating them
with examples, and we sketch some applications in number theory
Explicit formula for the generating series of diagonal 3D rook paths
Let denote the number of ways in which a chess rook can move from a
corner cell to the opposite corner cell of an
three-dimensional chessboard, assuming that the piece moves closer to the goal
cell at each step. We describe the computer-driven \emph{discovery and proof}
of the fact that the generating series admits
the following explicit expression in terms of a Gaussian hypergeometric
function: G(x) = 1 + 6 \cdot \int_0^x \frac{\,\pFq21{1/3}{2/3}{2} {\frac{27
w(2-3w)}{(1-4w)^3}}}{(1-4w)(1-64w)} \, dw.Comment: To appear in "S\'eminaire Lotharingien de Combinatoire
REDUCE package for the indefinite and definite summation
This article describes the REDUCE package ZEILBERG implemented by Gregor
St\"olting and the author.
The REDUCE package ZEILBERG is a careful implementation of the Gosper and
Zeilberger algorithms for indefinite, and definite summation of hypergeometric
terms, respectively. An expression is called a {\sl hypergeometric term}
(or {\sl closed form}), if is a rational function with respect
to . Typical hypergeometric terms are ratios of products of powers,
factorials, function terms, binomial coefficients, and shifted
factorials (Pochhammer symbols) that are integer-linear in their arguments
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