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...

The Riemann-Roch theorem on a graph G is closely related to Alexander duality in combinatorial commutive algebra. We study the lattice ideal given by chip firing on G and the initial ideal whose standard monomials are the G-parking functions. When G is a saturated graph, these ideals are generic and the Scarf complex is a minimal free resolution. Otherwise, syzygies are obtained by degeneration...

Koszul homology of monomial ideals provides a description of the structure of such ideals, not only from a homological point of view (free resolutions, Betti numbers, Hilbert series) but also from an algebraic viewpoint. In this paper we show that, in particular, the homology at degree (n− 1), with n the number of indeterminates of the ring, plays an important role for this algebraic descriptio...

We introduce the notion of Bernstein-Sato polynomial of an arbitrary variety, using the theory of V -filtrations of M. Kashiwara and B. Malgrange. We prove that the decreasing filtration of multiplier ideals coincides essentially with the restriction of the V -filtration. This implies a relation between the roots of the Bernstein-Sato polynomial and the jumping coefficients of the multiplier id...

The theory of “subalgebra basis” analogous to standard basis (the generalization of Gröbner bases to monomial ordering which are not necessarily well ordering [1].) for ideals in polynomial rings over a field is developed. We call these bases “SASBI Basis” for “Subalgebra Analogue to Standard Basis for Ideals”. The case of global orderings, here they are called “SAGBI Basis” for “Subalgebra Ana...

The construction of joins and secant varieties is studied in the combinatorial context of monomial ideals. For ideals generated by quadratic monomials, the generators of the secant ideals are obstructions to graph colorings, and this leads to a commutative algebra version of the Strong Perfect Graph Theorem. Given any projective variety and any term order, we explore whether the initial ideal o...

We introduce the notion of Bernstein-Sato polynomial of an arbitrary variety (which is not necessarily reduced nor irreducible), using the theory of V -filtrations of M. Kashiwara and B. Malgrange. We prove that the decreasing filtration by multiplier ideals coincides essentially with the restriction of the V -filtration. This implies a relation between the roots of the Bernstein-Sato polynomia...

We study the family of ideals defined by mixed size minors of two-sided ladders of indeterminates. We compute their Gröbner bases with respect to a skewdiagonal monomial order, then we use them to compute the height of the ideals. We show that these ideals correspond to a family of irreducible projective varieties, that we call mixed ladder determinantal varieties. We show that these varieties ...

We introduce the notion of Bernstein-Sato polynomial of an arbitrary variety, using the theory of V -filtrations of M. Kashiwara and B. Malgrange. We prove that the decreasing filtration of multiplier ideals coincides essentially with the restriction of the V -filtration. This implies a relation between the roots of the Bernstein-Sato polynomial and the jumping coefficients of the multiplier id...

We introduce the notion of Bernstein-Sato polynomial of an arbitrary variety, using the theory of V -filtrations of M. Kashiwara and B. Malgrange. We prove that the decreasing filtration of multiplier ideals coincides essentially with the restriction of the V -filtration. This implies a relation between the roots of the Bernstein-Sato polynomial and the jumping coefficients of the multiplier id...