Counting Lattice Points of Rational Polyhedra
نویسندگان
چکیده
منابع مشابه
Counting Lattice Points of Rational Polyhedra
The generating function F (P ) = ∑ α∈P∩ZN xα for a rational polytope P carries all essential information of P . In this paper we show that for any positive integer n, the generating function F (P, n) of nP = {nx : x ∈ P} can be written as F (P, n) = ∑ α∈A Pα(n)x, where A is the set of all vertices of P and each Pα(n) is a certain periodic function of n. The Ehrhart reciprocity law follows autom...
متن کاملCounting Lattice Points in Polyhedra
We present Barvinok’s 1994 and 1999 algorithms for counting lattice points in polyhedra. 1. The 1994 algorithm In [2], Barvinok presents an algorithm that, for a fixed dimension d, calculates the number of integer points in a rational polyhedron. It is shown in [6] and [7] that the question can be reduced to counting the number of integer points in a k-dimensional simplex with integer vertices ...
متن کاملSimple Explicit Formula for Counting Lattice Points of Polyhedra
Given z ∈ C and A ∈ Z, we consider the problem of evaluating the counting function h(y; z) := P { z |x∈Z;Ax=y, x≥0}. We provide an explicit expression for h(y; z) as well as an algorithm with possibly numerous but very simple calculations. In addition, we exhibit finitely many fixed convex cones of R explicitly and exclusively defined by A such that for any y ∈ Z, the sum h(y; z) can be obtaine...
متن کاملCounting Rational Points on Hypersurfaces
For any n ≥ 2, let F ∈ Z[x1, . . . , xn] be a form of degree d ≥ 2, which produces a geometrically irreducible hypersurface in P. This paper is concerned with the number N(F ; B) of rational points on F = 0 which have height at most B. For any ε > 0 we establish the estimate N(F ; B) = O(B), whenever either n ≤ 5 or the hypersurface is not a union of lines. Here the implied constant depends at ...
متن کاملTwo-lattice polyhedra: duality and extreme points
Two-lattice polyhedra are a special class of lattice polyhedra that include network 4ow polyhedra, fractional matching polyhedra, matroid intersection polyhedra, the intersection of two polymatroids, etc. In this paper we show that the maximum sum of components of a vector in a 2-lattice polyhedron is equal to the minimum capacity of a cover for the polyhedron. For special classes of 2-lattice ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Advances in Mathematics
سال: 2000
ISSN: 0001-8708
DOI: 10.1006/aima.2000.1931