In mathematics, an integral polytope has an associated Ehrhart polynomial that encodes the relationship between the volume of a polytope and the number of integer points the polytope contains. The theory of Ehrhart polynomials can be seen as a higher-dimensional generalization of Pick's theorem in the Euclidean plane.
These polynomials are named after Eugène Ehrhart who studied them in the 1960s.
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In mathematics, an integral polytope has an associated Ehrhartpolynomial that encodes the relationship between the volume of a polytope and the number...
≥ 13, the coefficient of t in the Ehrhartpolynomial of Tr is negative. This example shows that Ehrhartpolynomials can sometimes have negative coefficients...
applying a randomized polynomial-time approximation scheme for polytope volume. The Ehrhartpolynomial of the order polytope is a polynomial whose values at...
easily be computed from the leading coefficient of the Ehrhartpolynomial. The Ehrhartpolynomial associated with the Birkhoff polytope is only known for...
Ehrhart is a surname. Notable people with the surname include: Eugène Ehrhart (1906–2000), French mathematician who introduced Ehrhartpolynomials in the...
polyhedra are formalized by the Ehrhartpolynomials. These differ from figurate numbers in that, for Ehrhartpolynomials, the points are always arranged...
chromatic polynomial and Ehrhartpolynomial (see below), all special cases of Stanley's general Reciprocity Theorem. The chromatic polynomial P ( G , n...
coordinates is called a lattice polyhedron or integral polyhedron. The Ehrhartpolynomial of a lattice polyhedron counts how many points with integer coordinates...
research mathematics, figurate numbers are studied by way of the Ehrhartpolynomials, polynomials that count the number of integer points in a polygon or polyhedron...
polytope, including its volume and number of vertices, is encoded by its Ehrhartpolynomial. Integral polytopes are prominently featured in the theory of toric...
lattice is described by the polytope's Ehrhartpolynomial. Formulas for some of the coefficients of this polynomial involve d(Λ) as well. In certain approaches...
Dehn–Sommerville equations relating numbers of faces; Pick's theorem and the Ehrhartpolynomials, both of which relate lattice counting to volume; generating functions...
description) to combinatorial or algebraic properties (e.g., H-vector, Ehrhartpolynomial, Hilbert basis, and Schlegel diagrams). There are also many visualization...
{\displaystyle d} . Convex cone Algebraic geometry Number theory Ring theory Ehrhartpolynomial Rational cone Toric variety Stanley, Richard P. (1986). "Two poset...
into orthoschemes – is it possible for simplices of every dimension? Ehrhart's volume conjecture: a convex body K {\displaystyle K} in n {\displaystyle...
_{n})={\frac {(n+1)^{n}}{n!}}\sim {\frac {e^{n+1}}{\sqrt {2\pi n}}}.} Ehrhart's volume conjecture is that this is the (optimal) upper bound on the volume...