31 research outputs found
The equivariant Todd genus of a complete toric variety, with Danilov condition
We write the equivariant Todd class of a general complete toric variety as an
explicit combination of the orbit closures, the coefficients being analytic
functions on the Lie algebra of the torus which satisfy Danilov's requirement
Local Euler-Maclaurin formula for polytopes
We give a local Euler-Maclaurin formula for rational convex polytopes in a
rational euclidean space . For every affine rational polyhedral cone C in a
rational euclidean space W, we construct a differential operator of infinite
order D(C) on W with constant rational coefficients, which is unchanged when C
is translated by an integral vector. Then for every convex rational polytope P
in a rational euclidean space V and every polynomial function f (x) on V, the
sum of the values of f(x) at the integral points of P is equal to the sum, for
all faces F of P, of the integral over F of the function D(N(F)).f, where we
denote by N(F) the normal cone to P along F.Comment: Revised version (July 2006) has some changes of notation and
references adde
How to Integrate a Polynomial over a Simplex
This paper settles the computational complexity of the problem of integrating
a polynomial function f over a rational simplex. We prove that the problem is
NP-hard for arbitrary polynomials via a generalization of a theorem of Motzkin
and Straus. On the other hand, if the polynomial depends only on a fixed number
of variables, while its degree and the dimension of the simplex are allowed to
vary, we prove that integration can be done in polynomial time. As a
consequence, for polynomials of fixed total degree, there is a polynomial time
algorithm as well. We conclude the article with extensions to other polytopes,
discussion of other available methods and experimental results.Comment: Tables added with new experimental results. References adde
Summing a polynomial function over integral points of a polygon. User's guide
This document is a companion for the Maple program \textbf{Summing a
polynomial function over integral points of a polygon}. It contains two parts.
First, we see what this programs does. In the second part, we briefly recall
the mathematical background
Computation of the highest coefficients of weighted Ehrhart quasi-polynomials of rational polyhedra
This article concerns the computational problem of counting the lattice
points inside convex polytopes, when each point must be counted with a weight
associated to it. We describe an efficient algorithm for computing the highest
degree coefficients of the weighted Ehrhart quasi-polynomial for a rational
simple polytope in varying dimension, when the weights of the lattice points
are given by a polynomial function h. Our technique is based on a refinement of
an algorithm of A. Barvinok [Computing the Ehrhart quasi-polynomial of a
rational simplex, Math. Comp. 75 (2006), pp. 1449--1466] in the unweighted case
(i.e., h = 1). In contrast to Barvinok's method, our method is local, obtains
an approximation on the level of generating functions, handles the general
weighted case, and provides the coefficients in closed form as step polynomials
of the dilation. To demonstrate the practicality of our approach we report on
computational experiments which show even our simple implementation can compete
with state of the art software.Comment: 34 pages, 2 figure
Coefficients of Sylvester's Denumerant
For a given sequence of positive integers, we consider
the combinatorial function that counts the nonnegative
integer solutions of the equation , where the right-hand side is a varying
nonnegative integer. It is well-known that is a
quasi-polynomial function in the variable of degree . In combinatorial
number theory this function is known as Sylvester's denumerant.
Our main result is a new algorithm that, for every fixed number , computes
in polynomial time the highest coefficients of the quasi-polynomial
as step polynomials of (a simpler and more explicit
representation). Our algorithm is a consequence of a nice poset structure on
the poles of the associated rational generating function for
and the geometric reinterpretation of some rational
generating functions in terms of lattice points in polyhedral cones. Our
algorithm also uses Barvinok's fundamental fast decomposition of a polyhedral
cone into unimodular cones. This paper also presents a simple algorithm to
predict the first non-constant coefficient and concludes with a report of
several computational experiments using an implementation of our algorithm in
LattE integrale. We compare it with various Maple programs for partial or full
computation of the denumerant.Comment: minor revision, 28 page
INTERMEDIATE SUMS ON POLYHEDRA II:BIDEGREE AND POISSON FORMULA
Abstract. We continue our study of intermediate sums over polyhedra,interpolating between integrals and discrete sums, whichwere introduced by A. Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), 1449–1466]. By well-known decompositions, it is sufficient to considerthe case of affine cones s+c, where s is an arbitrary real vertex andc is a rational polyhedral cone. For a given rational subspace L,we integrate a given polynomial function h over all lattice slicesof the affine cone s + c parallel to the subspace L and sum up theintegrals. We study these intermediate sums by means of the intermediategenerating functions SL(s+c)(ξ), and expose the bidegreestructure in parameters s and ξ, which was implicitly used in thealgorithms in our papers [Computation of the highest coefficients ofweighted Ehrhart quasi-polynomials of rational polyhedra, Found.Comput. Math. 12 (2012), 435–469] and [Intermediate sums onpolyhedra: Computation and real Ehrhart theory, Mathematika 59(2013), 1–22]. The bidegree structure is key to a new proof for theBaldoni–Berline–Vergne approximation theorem for discrete generatingfunctions [Local Euler–Maclaurin expansion of Barvinokvaluations and Ehrhart coefficients of rational polytopes, Contemp.Math. 452 (2008), 15–33], using the Fourier analysis with respectto the parameter s and a continuity argument. Our study alsoenables a forthcoming paper, in which we study intermediate sumsover multi-parameter families of polytopes
Three Ehrhart Quasi-polynomials
Let be a semi-rational parametric polytope, where
is a real multi-parameter. We study intermediate sums of
polynomial functions on , where we
integrate over the intersections of with the subspaces parallel to a
fixed rational subspace through all lattice points, and sum the integrals.
The purely discrete sum is of course a particular case (), so counts the integer points in the parametric polytopes.
The chambers are the open conical subsets of such that the shape of
does not change when runs over a chamber. We first prove that on
every chamber of , is given by a quasi-polynomial function
of . A key point of our paper is an analysis of the interplay between
two notions of degree on quasi-polynomials: the usual polynomial degree and a
filtration, called the local degree.
Then, for a fixed , we consider a particular linear combination of
such intermediate weighted sums, which was introduced by Barvinok in order to
compute efficiently the highest coefficients of the Ehrhart
quasi-polynomial which gives the number of points of a dilated rational
polytope. Thus, for each chamber, we obtain a quasi-polynomial function of ,
which we call Barvinok's patched quasi-polynomial (at codimension level ).
Finally, for each chamber, we introduce a new quasi-polynomial function of
, the cone-by-cone patched quasi-polynomial (at codimension level ),
defined in a refined way by linear combinations of intermediate generating
functions for the cones at vertices of .
We prove that both patched quasi-polynomials agree with the discrete weighted
sum in the terms corresponding to the highest
polynomial degrees.Comment: 41 pages, 13 figures; v2: changes to introduction, new graphics; v3:
add more detailed references, move example to introduction; v4: fix
reference
Top Coefficients of the Denumerant
International audienceFor a given sequence of positive integers, we consider the combinatorial function that counts the nonnegative integer solutions of the equation , where the right-hand side is a varying nonnegative integer. It is well-known that is a quasipolynomial function of of degree . In combinatorial number theory this function is known as the . Our main result is a new algorithm that, for every fixed number , computes in polynomial time the highest coefficients of the quasi-polynomial as step polynomials of . Our algorithm is a consequence of a nice poset structure on the poles of the associated rational generating function for and the geometric reinterpretation of some rational generating functions in terms of lattice points in polyhedral cones. Experiments using a implementation will be posted separately.Considérons une liste de entiers positifs. Le dénumérant est lafonction qui compte le nombre de solutions en entiers positifs ou nuls de l’équation , où varie dans les entiers positifs ou nuls. Il est bien connu que cette fonction est une fonction quasi-polynomiale de , de degré . Nous donnons un nouvel algorithme qui calcule, pour chaque entier fixé (mais n’est pas fixé, les plus hauts coefficients du quasi-polynôme en termes de fonctions en dents de scie. Notre algorithme utilise la structure d’ensemble partiellement ordonné des pôles de la fonction génératrice de . Les plus hauts coefficients se calculent à l’aide de fonctions génératrices de points entiers dans des cônes polyèdraux de dimension inférieure ou égale Ã
Quantum cohomology of partial flag manifolds
We compute the quantum cohomology rings of the partial flag manifolds
F_{n_1\cdots n_k}=U(n)/(U(n_1)\times \cdots \times U(n_k)). The inductive
computation uses the idea of Givental and Kim. Also we define a notion of the
vertical quantum cohomology ring of the algebraic bundle. For the flag bundle
F_{n_1\cdots n_k}(E) associated with the vector bundle E this ring is found.Comment: 33 page