The regularity of a toric variety

We give a method for computing the degrees of the minimal syzygies of a toric variety by means of combinatorial techniques. Indeed, we complete the explicit description of the minimal free resolution of the associated semigroup algebra, using the simplicial representation of Koszul homology which appeared in A. Campillo and C. Marijuán (1991, Sém. Théor. Nombres Bordeaux3, 249–260). As an application, we obtain an algorithm for computing the Castelnuovo–Mumford regularity of a projective toric variety. This regularity is explicitly bounded by means of the semigroup generators which parametrize the variety

Preuve automatique dans le calcul propositionnel et les logiques trivalentes

Nous présentons une application des bases de Gröbner (bases standard) d’idéaux de polynômes la vérification des tautologies dans le Calcul Propositionnel et dans trois types de logiques trivalentes. L’idée base est de transformer les formules en polynômes, et de trouver l’´equivalent algèbrique de la déduction: nous verons des théorèmes faisant la liaison entre déduction et crièeres de réduction algèbrique

Minimal resolutions of lattice ideals and integer linear programming

A combinatorial description of the minimal free resolution of a lattice ideal allows us to the connection of Integer Linear Programming and Algebra. The non null reduced homology spaces of some simplicial complexes are the key. The extremal rays of the associated cone reduce the number of variables.Ministerio de Ciencia y TecnologíaJunta de Andalucí

Combinatorics of syzygies for semigroup algebras

We describe how the graded minimal resolution of certain semigroup algebras is related to the combinatorics of some simplicial complexes. We obtain characterizations of the Cohen-Macaulay and Gorenstein conditions. The Cohen-Macaulay type is computed from combinatorics. As an application, we compute explicitly the graded minimal resolution of monomial both affine and simplicial projective surfaces.Dirección General de Investigación Científica y TécnicaJunta de Andalucí