The well known correspondence between even cycles of an undirected graph and polynomials in a binomial ideal associated to a graph is extended to odd cycles and polynomials in another binomial ideal. Other binomial ideals associated to an undirected graph are also introduced. The results about them with topics on monomial ideals are used in order to show decision procedures for bipartite graphs...

where lm(I) denotes the ideal generated by the leading monomials of the elements of I. This condition has already been studied in Bayer et al. (1991) and it has been shown that (1.1) holds for any ideal and any term order if and only if π is flat. In this paper we study condition (1.1) under the additional assumption that R′ is not a general Noetherian commutative ring with identity but a field...

The distortion varieties of a given projective variety are parametrized by duplicating coordinates and multiplying them with monomials. We study their degrees and defining equations. Exact formulas are obtained for the case of one-parameter distortions. These are based on Chow polytopes and Gröbner bases. Multi-parameter distortions are studied using tropical geometry. The motivation for distor...

We study the structure of a family of algebras which encodes a generalization of the Pieri Rule for the complex orthogonal group. In particular, we show that each of these algebras has a standard monomial basis and has a flat deformation to a Hibi algebra. There is also a parallel theory for the complex symplectic group.

Generic linkage is used to compute a prime ideal such that the radical of the initial ideal of the prime ideal is equal to the radical of a given codimension two monomial ideal that has a Cohen-Macaulay quotient ring.

Gröbner basis techniques in the algebraic study of triangulations of convex polytopes as well as of the number of faces of simplicial complexes will be discussed. Of these two traditional topics in combinatorics, the first will be studied by using initial ideals of toric ideals and the second will be studied by using generic initial ideals of monomial ideals.

In this paper, we study the basic problem of counting independent sets in a graph and, in particular, the problem of counting antichains in a finite poset, from an algebraic perspective. We show that neither independence polynomials of bipartite Cohen-Macaulay graphs nor Hilbert series of initial ideals of radical zero-dimensional complete intersections ideals, can be evaluated in polynomial ti...

We establish some results in the theory of standard bases of differential ideals (also known as differential Gröbner bases) in differential polynomial rings. This work was stimulated by some articles of V. Weispfenning, devoted to the classification of term-orders and parametric Gröbner bases of ideals generated by monomials and binomials.

In this paper, a new  algorithm for computing secondary invariants of  invariant rings of monomial groups is presented. The main idea is to compute simultaneously a truncated SAGBI-G basis and the standard invariants of the ideal generated by the set of primary invariants.  The advantage of the presented algorithm lies in the fact that it is well-suited to complexity analysis and very easy to i...

In this paper, we develop the Gr6bner-Shirshov basis theory for the representations of associative algebras by introducing the notion of Gr6bner-Shirshov pairs. Our result can be applied to solve the reduction problem in representation theory and to construct monomial bases of representations of associative algebras. As an illustration, we give an explicit construction of Gr6bner-Shirshov pairs...