Abdallah Assi

The Gröbner fan of an An-module

Abstract Let I be a non-zero left ideal of the Weyl algebra A n of order n over a field k and let L: R 2n → R be a linear form defined by L(α,β)=∑ i=1 n e i α i +∑ i=1 n f i β i . If e i +f i ≥0 , then L defines a filtration F • L on A n . Let gr L (I) be the graded ideal associated with the filtration induced by F • L on I . Let finally U denote the set of all linear form L for which e i +f i ≥0 for all 1≤i≤n . The aim of this paper is to study, by using the theory of Grobner bases, the stability of gr L (I) when L varies in U . In a previous paper, we obtained finiteness results for some particular linear forms (used in order to study the regularity of a D -module along a smooth hypersurface). Here we generalize these results by adapting the theory of Grobner fan of Mora-Robbiano to the D -module case. Our main tool is the homogenization technique initiated in our previous paper, and recently clarified in a work by F. Castro-Jimenez and L. Narvaez-Macarro.

NUMERICAL SEMIGROUPS AND APPLICATIONS

The aim of this manuscript is to give some basic notions related to numerical semigroups, and from these on the one hand describe a classical application to the study of singularities of plane algebraic curves, and on the other, show how numerical semigroups can be used to obtain handy examples of nonunique factorization invariants.

On flatness of generic projections

Abstract Let R be a commutative noetherian ring and let I be an ideal of R [ x 1 ,…, x n = R [ x ]. The morphism ψ R → R [ x ]/ I defines a family of algebraic varieties as follows: Let p be a prime ideal of R (or an element of Spec R ) and let K ( p ) be the quotient field of the localization R p of R at p , then we have an algebraic variety in A n K ( p ) defined by K ( p )[ x ]/ I ( p ) where I ( p ) = I.K ( p )[ x ]. When p varies, these varieties are called fibers of ψ. On the other hand, when ψ is flat, many properties are preserved in the fibers. The main objective of this paper is to characterize flatness of ψ by studying the relationship with the notions of Grobner and standard bases. When R is principal, we obtain an algorithm to compute the maximal generic open set of flatness of Spec R and then we give some applications related to this situation.

Constructing the set of complete intersection numerical semigroups with a given Frobenius number

Delorme suggested that the set of all complete intersection numerical semigroups can be computed recursively. We have implemented this algorithm, and particularized it to several subfamilies of this class of numerical semigroups: free and telescopic numerical semigroups, and numerical semigroups associated to an irreducible plane curve singularity. The recursive nature of this procedure allows us to give bounds for the embedding dimension and for the minimal generators of a semigroup in any of these families.

Bases of subalgebras of K〚x〛 and K[x]

Let

Algorithms for curves with one place at infinity

f\_1,\ldots, f\_s

Bases of subalgebras of K ź x ź and K x

be formal power series (respectively polynomials) in thevariable

Bases of subalgebras of K〚x〛K〚x〛 and K[x]K[x]

x

Semigroup of an Irreducible Meromorphic Series

. We study the semigroup of orders of the formal series inthe algebra

Numerical Semigroups, the Basics

K[[ f1,\ldots, f\_s]] \subseteq K[[ x ]]