Jose Maria Ucha-Enríquez

Explicit Comparison Theorems for D -modules

We prove in an explicit way a duality formula between two A2-modulesMlog andMlog associated to a plane curve and we give an application of this duality to the comparison between Mlogand theA2 -module of rational functions along the curve. We treat the analytic case as well.

Testing the Logarithmic Comparison Theorem for Free Divisors

We propose in this work a computational criterion to test if a free divisor D ⊂ Cn verifies the Logarithmic Comparison Theorem (LCT); that is, whether the complex of logarithmic differential forms computes the cohomology of the complement of D in Cn . For Spencer free divisors D ≡ (f = 0), we solve a conjecture about the generators of the annihilating ideal of 1/f and make a conjecture on the nature of Euler homogeneous free divisors which verify LCT. In addition, we provide examples of free divisors defined by weighted homogeneous polynomials that are not locally quasi-homogeneous.

Logarithmic Comparison Theorem and some Euler homogeneous free divisors

Let D, x be a free divisor germ in a complex manifold X of dimension n > 2. It is an open problem to find out which are the properties required for D,x to satisfy the so-called Logarithmic Comparison Theorem (LCT), that is, when the complex of logarithmic differential forms computes the cohomology of the complement of D,x. We give a family of Euler homogeneous free divisors which, somewhat unexpectedly, does not satisfy the LCT.

Sudokus and Gröbner bases: not only a divertimento

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 Grobner 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.

Nouvelle Cuisine for the computation of the annihilating ideal of f /s

Let f1,..., fp be polynomials in C[x1,..., xn] and let D = Dn be the n-th Weyl algebra. The annihilating ideal of

Comparison of theoretical complexities of two methods for computing annihilating ideals of polynomials

f^{s}=f_{1}^{s1}...f_{p}^{sp}

Gröbner bases and logarithmic D-modules

in D[s]=D[s1,...,sp] is a necessary step for the computation of the Bernstein-Sato ideals of f1,..., fp. We point out experimental differences among the efficiency of the available methods to obtain this annihilating ideal and provide some upper bounds for the complexity of its computation.