Gröbner bases of lattices, corner polyhedra, and integer programming
There are very close connections between the arithmetic of integer lattices, algebraic properties of the associated ideals, and the geometry and the combinatorics of corresponding polyhedra. In this paper, we investigate the generating sets (``Gröbner bases'') of integer lattices that correspond to the Gröbner bases of the associated binomial ideals. Extending results by Sturmfels\&Thomas, we obtain a geometric characterization of the universal Gröbner basis in terms of the vertices and edges of the associated corner polyhedra. In the special case where the lattice has finite index, the corner polyhedra were studied by Gomory, and there is a close connection to the ``group problem in integer programming''. We present exponential lower and upper bounds for the maximal size of a reduced Gröbner basis. The initial complex of (the ideal of) a lattice is shown to be dual to the boundary of a certain simple polyhedron.
scientific article; zbMATH DE number 821696