Integrals and Sums over rational polytopes
This webpage hosts papers and
softwares about the problem of computing integrals and sums of
polynomial functions over rational polyhedra. In particular it computes the volume and the number of
integral points in rational polyhedra (and its applications to
representation theory).
Link to LattE software
- How to Integrate a Polynomial over a Simplex
Authors: M. W. Baldoni, N. Berline, ,
J. De Loera, ,
Matthias Koëppe, M. Vergne.
Paper: pdf
(365 ko).
Status of the paper:
Published in Mathematics of Computation, vol.80, no. 273 (2010), p. 297-325.
Also available on the arXiv : math/0809.2083
Software: LattE Integrale
- Summing a polynomial
function over integral points of a polygon. User's guide
Authors: M. W. Baldoni, N. Berline, M. Vergne.
Paper: pdf
(161 ko).
Status of the
paper:
submitted. Also available on the arXiv : math/0905.1820
Softwares: compressed
archive (17 ko) containing Maple worksheets.
See also : Oberwolfach References on Mathematicals Software
- Volume computation for
polytopes and partition functions for classical root systems
Authors: M. W. Baldoni, M. Beck, C. Cochet, M. Vergne.
Abstract: 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.
Paper: pdf
(459 ko).
Status of the paper:
Published in Discrete and Computational Geometry 35, 551-595.
Also available on the arXiv : math/0504231
Softwares: compressed
archive (60 ko) containing Maple worksheets.
- Vector Partition and
Representation Theory
Author: C. Cochet.
Abstract:
We apply some recent developments of Baldoni-Beck-Cochet-Vergne on
vector partition function, to Kostant's and Steinberg's formulae, for
classical Lie algebras $A_r$, $B_r$, $C_r$, $D_r$. We therefore get
efficient {\tt Maple} programs that compute for these Lie algebras: the
multiplicity of a weight in an irreducible finite-dimensional
representation; the decomposition coefficients of the tensor product of
two irreducible finite-dimensional representations. These programs can
also calculate associated Ehrhart quasipolynomials.
Paper: ps
(2,6 Mo), pdf
(500 ko).
Book: Kotska Numbers and Littlewood-Richardson Coefficients
arXiv: math/0506159
Softwares:
compressed
archive (44 ko) containing Maple worksheets.
- Counting Integer Flows in
Networks
Authors: M. W. Baldoni,
J. De Loera, M. Vergne.
Abstract:
This paper discusses new analytic algorithms and software for the
enumeration of all integer flows inside a network. Concrete
applications abound in graph theory \cite{Jaeger}, representation
theory \cite{kirillov}, and statistics \cite{persi}. Our methods
clearly surpass traditional exhaustive enumeration and other
algorithms and can even yield formulas when the input data
contains some parameters. These methods are based on the study of
rational functions with poles on arrangements of hyperplanes.
Paper: ps
(500 ko), pdf
(400 ko).
Status of the paper:
Found. Comput. Math. 4 (2004), no.3, 277-314.
Also available on the arXiv : math/0303228
Softwares: compressed
archive (12 ko) containing Maple worksheets.