Let A n (k) be the Weyl algebra, with k a field of characteristic zero. It is known that every projective finitely generated left module is free or isomorphic to a left ideal. Let M be a left submodule of a free module. In this paper we give an algorithm to compute the projective dimension of M. If M is projective and rank(M) ≥ 2 we give a procedure to find a basis.

Abstract. Sudoku is a logic-based placement puzzle. We recall how to translate this puzzle into a 9-colouring problem which is equivalent to a (big) algebraic system of polynomial equations. We study how far Gröbner bases techniques can be used to treat these systems produced by Sudokus. This general purpose tool can not be considered as a good solver, but we show that it can be useful to provide information on systems that are -in spite of their origin-hard to solve.

Let f1, . . . , fp be polynomials in C[x1, . . . , xn] and let D = Dn be the n-th Weyl algebra. We provide upper bounds for the complexity of computing the annihilating ideal of f s = f s1 1 · · · f sp p in D[s] = D[s1, . . . , sp]. These bounds provide an initial explanation on the differences between the running times of the two methods known to obtain the so-called BernsteinSato ideals.Ministerio de Ciencia y Tecnología MTM2004-01165Junta de Andalucía FQM-33

A classical result in K-Theory about polynomial rings like the Quillen-Suslin theorem admits an algorithmic approach when the ring of coefficients has some computational properties, associated with Gröbner bases. There are several algorithms when we work in K[x 1 , . . . , xn], K a field. In this paper we compute a free basis of a finitely generated projective module over R[x 1 , . . . , xn], R a principal ideal domain with additional properties, test the freeness for projective modules over D[x 1 , . . . , xn], with D a Dedekind domain like Z[ √ −5] and for the one variable case compute a free basis if there exists any.

Abstract. Many problems in engineering design involve the use of nonlinearities and some integer variables. Methods based on test sets have been proposed to solve some particular problems with integer variables, but they have not been frequently applied because of computation costs. The walk-back procedure based on a test set gives an exact method to obtain an optimal point of an integer programming problem with linear and nonlinear constraints, but the calculation of this test set and the identification of an optimal solution using the test set directions are usually computationally intensive.In problems for which obtaining the test set is reasonably fast, we show how the effectiveness can still be substantially improved. This methodology is presented in its full generality and illustrated on two specific problems: (1) minimizing cost in the problem of scheduling jobs on parallel machines given restrictions on demands and capacity, and (2) minimizing cost in the series parallel redundancy allocation problem, given a target reliability. Our computational results are promising and suggest the applicability of this approach to deal with other problems with similar characteristics or to combine it with mainstream solvers to certify optimality.Non-linear Integer Programming and test set and Gröbner basis and chance constrained programming