Ehrhart

Ehrhart is a surname of French origin. In mathematics, it is most commonly associated with Eugène Ehrhart, a French mathematician who introduced the theory of counting lattice points in dilations of lattice polytopes, now called Ehrhart theory.

Key result, Ehrhart's theorem, concerns a d-dimensional lattice polytope P (a convex polytope with vertices in Z^d). The function L_P(n) = the number of lattice points in the dilate nP is a polynomial in n of degree d. The leading coefficient equals the d-dimensional volume (the usual Euclidean volume, normalized so that the unit lattice has volume 1) of P.

Ehrhart reciprocity states that the number of lattice points in the interior of nP equals (-1)^d L_P(-n).

For rational polytopes, the corresponding counting function is a quasi-polynomial rather than a polynomial; more generally, the Ehrhart series, a generating function sum_{n>=0} L_P(n) t^n, is a rational function with denominator (1-t)^{d+1}.

Applications of Ehrhart theory appear in combinatorics, number theory, and optimization, notably in enumerating lattice points in polytopes arising in integer programming and in the study of toric varieties in algebraic geometry. The theory has been extended to various generalizations, including weighted and colored lattice point counts and connections to Hilbert functions.

Outside mathematics, Ehrhart as a surname may refer to individuals bearing the name; the most prominent associated figure is Eugène Ehrhart.