Ehrhart polynomial

Ehrhart polynomial
Name	Ehrhart polynomial
Field	Mathematics
Introduced	1960s
Founder	Eugène Ehrhart

Ehrhart polynomial The Ehrhart polynomial is a function associated with a convex lattice polytope that counts integer lattice points in integral dilates; it connects discrete geometry, combinatorics, and number theory. Introduced by Eugène Ehrhart, it has deep ties to enumerative geometry, convex polytopes, and algebraic combinatorics, and interacts with topics ranging from Hilbert series to toric varieties.

Definition and basic properties

For a d-dimensional convex lattice polytope P (with vertices in the integer lattice Z^d), the Ehrhart polynomial L_P(n) gives the number of lattice points in the n-fold dilation nP for positive integer n. In Ehrhart’s theorem the counting function is a polynomial of degree d with leading coefficient equal to the Euclidean volume of P and with constant term 1. The coefficients and values of L_P(n) relate to the geometric structure of P and to invariants appearing in algebraic geometry such as Hilbert polynomials of projective toric varieties. Key people and contexts connected to these properties include Eugène Ehrhart, Pierre Deligne, Jean-Pierre Serre, David Mumford, and institutions like the Institut des Hautes Études Scientifiques and the Collège de France.

Ehrhart–Macdonald reciprocity

Ehrhart–Macdonald reciprocity states that the Ehrhart polynomial evaluated at negative integers counts (up to sign) the number of interior lattice points of nP; equivalently, L_P(-n) = (-1)^d L_{P^\circ}(n) where P^\circ denotes the interior. This reciprocity theorem has parallels with reciprocity in algebraic geometry explored by Alexander Grothendieck, Jean-Louis Koszul, and John Tate, and it echoes duality phenomena studied by Henri Poincaré and Alexander Grothendieck. Macdonald’s contribution linked the reciprocity to generating functions and to identities appearing in the work of Percy MacMahon and Srinivasa Ramanujan.

Computation and algorithms

Computing Ehrhart polynomials efficiently involves techniques from computational geometry and commutative algebra. Barvinok’s algorithm and subsequent improvements by Alexander Barvinok and Kevin Woods allow counting lattice points in fixed dimension using short rational generating functions and the theory of parametric integer programming developed by George Nemhauser and Laurence Wolsey. Software packages inspired by work at the Mathematical Sciences Research Institute and institutions like the Institut Camille Jordan implement algorithms using the Lenstra–Lenstra–Lovász lattice reduction algorithm and connections to Grӧbner bases studied by Bruno Buchberger. Complexity results link to the Cook–Levin theorem and to algorithmic studies by Richard Karp and Jack Edmonds.

Examples and special cases

Classic examples include unimodular simplices, standard cubes, and rational polytopes arising from root systems such as those studied by Wilhelm Killing and Élie Cartan. For a unit interval [0,1] the Ehrhart polynomial is linear; for the standard d-cube it equals (n+1)^d, linking to binomial coefficient identities investigated by James Joseph Sylvester and Hermann Minkowski’s lattice point theorems. Reflexive polytopes appearing in mirror symmetry studied by Victor Batyrev and Maxim Kontsevich provide notable special cases where duality and symmetry constraints determine Ehrhart coefficients, and examples from optimization include Birkhoff polytopes connected to George Dantzig and Richard Brualdi.

Connections to geometry and combinatorics

Ehrhart polynomials serve as a bridge between convex geometry, toric geometry, and enumerative combinatorics. They appear in the study of toric varieties developed by David Cox and William Fulton, in the theory of Newton polytopes related to Isaac Newton and Bernhard Riemann’s influence on algebraic curves, and in Stanley’s work relating order polytopes and posets to generating functions. Connections extend to the study of zonotopes explored by Hermann Minkowski, to matroid polytopes linked to Tom Brylawski and Gian-Carlo Rota, and to partition theory developed by Srinivasa Ramanujan and Leonard Euler. Applications intersect with representation theory in contexts studied by Hermann Weyl and Harish-Chandra, and with statistical mechanics models where lattice-point enumeration echoes contributions by Lars Onsager.

Generalizations and extensions

Generalizations include Ehrhart quasipolynomials for rational polytopes, multivariate Ehrhart series tied to Hilbert series and Castelnuovo–Mumford regularity studied by Oscar Zariski and Jean-Pierre Serre, and weighted and equivariant Ehrhart theory connected to the work of Michael Atiyah and Isadore Singer on index theory. Further extensions relate to discrete tomography investigated by Gabor T. Herman, to valuation theory linked to Henri Lebesgue and P.D. Milman, and to tropical geometry advanced by Grigory Mikhalkin and Bernd Sturmfels. Active research directions involve mirror symmetry conjectures by Maxim Kontsevich, topological methods influenced by René Thom, and algorithmic improvements inspired by the European research community and global research centers such as the Isaac Newton Institute.

Category:Combinatorics