Study of universal Grobner bases of zero-dimensional lattice ideals

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The lattice ideal of dimension zero appears in the research of both pure mathematics and applied mathematics. The original purpose of the present research project was to establish the algebraic theory of universal Groebner bases of lattice ideal of dimension zero and to study of its theoretical effectivity to commutative algebra and algebraic geometry as well as its practical effectivity to integer programming, coding theory together with algebraic statistics. First, we investigated the universal Groebner basis of the lattice ideal of dimension zero arising from the toric ideal of a finite graph and succeeded in describing its structure in terms of the finite graph. Second, in the study of a problem on integer programming arising from a finite graph for which Gomory's relaxation can be applied, we developed the technique to decide the estimation of the computational complexity of finding an optimal solution by using combinatorics on finite graphs. Third, we achieved the study of finding an explicit expression of the corner polyhedron of a lattice ideal of dimension zero in terms of the Minkowski sum of a bounded convex polytope and a convex cone, and obtained some results on the combinatorics of the polyhedral structure of the corner polyhedron. Finally, we developed the algebraic study on the Markov basis of the contingency table in algebraic statistics and presented a statistic model arising from a complete multipartite graph. These research results will contribute to the development of the algebraic study of integer programming. In addition, we organized two international meetings related with computational commutative algebra and Groebner bases.