We study the problem of minimizing c \Delta x subject to A \Delta x = b, x 0 and x integral, for a fixed matrix A. Two cost functions c and c 0 are considered equivalent if they give the same optimal solutions for each b. We construct a polytope St(A) whose normal cones are the equivalence classes. Explicit inequality presentations of these cones are given by the reduced Gröbner bases associated with A. The union of the reduced Gröbner bases as c varies (called the universal Gröbner basis) consists precisely of the edge directions of St(A). We present geometric algorithms for computing St(A), the Graver basis [Gra], and the universal Gröbner basis.

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 Grobner basis. The initial complex of (the ideal of) a lattice is shown to be dual to the boundary of a certain simple polyhedron.

The set of all group relaxations of an integer program contains certain special members called Gomory relaxations. A family of integer programs with a fixed coefficient matrix and cost vector but varying right hand sides is a Gomory family if every program in the family can be solved by one of its Gomory relaxations. In this paper, we characterize Gomory families. Every TDI system gives a Gomory family, and we construct Gomory families from matrices whose columns form a Hilbert basis for the cone they generate. The existence of Gomory families is related to the Hilbert covering problems that arose from the conjectures of Sebö. Connections to commutative algebra are outlined at the end.

The algebraic technique of Gröbner bases is applied to study triangulations of the second hypersimplex \Delta(2; n). We present a quadratic Gröbner basis for the associated toric ideal I(Kn ). The simplices in the resulting triangulation of \Delta(2; n) have unit volume, and they are indexed by subgraphs which are linear thrackles [28] with respect to a circular embedding of Kn . For n 6 the number of distinct initial ideals of I(Kn ) exceeds the number of regular triangulations of \Delta(2; n); more precisely, the secondary polytope of \Delta(2; n) equals the state polytope of I(Kn ) for n 5 but not for n 6. We also construct a non-regular triangulation of \Delta(2; n) for n 9. We determine an explicit universal Gröbner basis of I(Kn ) for n 8. Potential applications in combinatorial optimization and random generation of graphs are indicated.

The toric ideal I of a matrix A"(a,...,a)3� � � � � ��is the kernel of the monoid algebra map π ˆ : k[x,...,x]Pk[t � � � $ �,...,t � $ �], defined as x>t��. It was shown in [4] that the reduced Gro ¨ bner basis of I, with respect to the weight vector c, can be used to solve all integer programs minimize �cx: Ax"b, x3���, denoted IP,asbvaries. In this paper we describe the construction of a truncated Gröbner basis of I with respect to c, that solves IP for a fixed b. This is achieved by establishing the homogeneity of I with respect to a multivariate grading induced by A. Depending on b, the truncated Gröbner basis may be considerably smaller than the entire Gröbner basis of I with respect to c. For programs of the form maximize�cx: Ax6b, x6u, x3 � � � in which all data are non-negative, this algebraic method gives rise to a combinatorial algorithm presented in [17].

This article is a brief survey of recent work on Gröbner bases (Buchberger 1965) of toric ideals and their role in integer programming. Toric varieties and ideals are crucial players in the interaction between combinatorics, discrete geometry, commutative algebra and algebraic geometry. For a detailed treatment of this topic see (Sturmfels 1995). Our survey focuses on the application of toric ideals to integer programming, a specific branch of discrete optimization, and for the sake of brevity we leave details to the references that are included. We study a family of integer programs associated with a fixed matrix A

this paper we will discuss computational and structural properties of subalgebras of polynomial rings when the base ring is a principal ideal domain (PID). The objects we study are the so-called SAGBI (subalgebra analogues of Grobner bases for ideals) bases for the subalgebras themselves and SAGBI-Grobner bases for the ideals in the subalgebras (SG bases). We will discuss how to compute these objects, and our goal is to avoid computations over the PID as much as possible. Further we will show the existence of strong SAGBI bases for these subalgebras and give an algorithm to compute them. For the general theory of SAGBI and SAGBI-Grobner bases over any commutative Noetherian ring we refer the reader to Miller [7]. In [7] algorithms are given for the computation of SAGBI and SG bases over an arbitrary Noetherian commutative ring R. In addition to the usual Buchberger-style algorithms the algorithms presented there relied on elimination order computations of Grobner bases over R. When R is a field, these extra Grobner basis computations were replaced by computing the minimal Hilbert basis for the set of solutions of certain linear diophantine equations. These in turn can be constructed by Grobner basis techniques, but over a field. In this paper first we show that in the construction of SAGBI bases over R, a PID, we can avoid these extra Grobner basis computations over R. Next we go on to consider the same question for SG bases. Here we show that the elimination order computations over R can be replaced by a degree reverse lexicographic (degrevlex) computation over the same ring R. In the last section we will show that strong SAGBI bases, the analogue of strong Grobner bases for ideals in polynomial rings, always exist, and we will give an algorithm for their construction....

Abstract. The toric Hilbert scheme of a lattice L ⊆ Z n is the multigraded Hilbert scheme parameterizing all ideals in k[x1,..., xn] with Hilbert function value one for every g in the grading monoid G + = N n /L. In this paper we show that if L is twodimensional, then the toric Hilbert scheme of L is smooth and irreducible. This result is false for lattices of dimension three and higher as the toric Hilbert scheme of a rank three lattice can be reducible. 1.

This paper is about algorithmic invariant theory as it is required within equivariant

Abstract not found