Alessandro Logar

Algorithms for the Quillen-Suslin theorem

Abstract We give an algorithm for computing a free basis of a projective C [x1,…xn,-module which is presented as the image, kernel, or cokernel of a polynomial matrix. Our method can be implemented using Grobner bases, and it provides an elementary, constructive proof of the Quillen-Suslin theorem.

Membership problem, Representation problem and the Computation of the Radical for one-dimensional Ideals

In this paper we consider membership problem, representation problem and also the computation of the radical for one-dimensional ideals in the polynomial ring k[X 1,…, X n ] from a complexity point of view. Our aim is to give bounds for the complexity of the above problems which are simply exponential in the number n of variables in the one-dimensional case. Moreover we show that in the general case the first two problems are doubly exponential only in the dimension of the ideal, and parallelizable. Many authors considered membership problem (i.e. given polynomials F, F 1,…, F, ∈ k[X 1,…, X n ], decide whether F belongs to I:= (F1…, F,)) and representation problem (compute a representation A i ∈ k[X 1,…, X n ]) from this effective approach. In particular [18] gives a lower bound, doubly exponential in (a fraction of) the numbers of variables n. On the other hand, [4] shows that in “good” cases, for instance for unmixed ideals, the membership problem is simply exponential in n.

Parametrization of the orbits of cubic surfaces

The aim of the paper is the study of the orbits of the action of PGL4 on the space ℙ3 of the cubic surfaces of ℙ3, i.e., the classification of cubic surfaces up to projective motions. A varietyQ⊂ℙ19 is explicitely constructed as the union of 22 disjoint irreducible components which are either points or open subsets of linear spaces. More precisely, each orbit of the above action intersects one componentX ofQ in a finite number of points and the action of PGL4 restricted on each componentX is equivalent to the action of a finite groupGX onX which can be explicitely computed. Finally the cubic surfaces of each component ofQ are studied in details by determining their stabilizers, their rational representations and whether they can be expressed as the determinant of a 3×3 matrix of linear forms.The results are obtained with computational techniques and with the aid of some computer algebra systems like CoCoA, Macaulay and Maple.

Monoidal closed structures on categories with constant maps

The purpose of this paper is to study the so-called canonical monoidal closed structures on concrete categories with constant maps. First of all we give an example of a category of this kind where there exists a non canonical monoidal closed structure. Later, we give a technique to construct a class of suitable full subcategories of the category of T 0 -spaces, such that all monoidal closed structures on them are canonical. Finally we show that “almost all” useful categories of topological compact spaces admit no monoidal closed structures whatsoever.

ACTION OF THE BOREL GROUP ON MONOMIAL IDEALS

ABSTRACT Let be the Borel group of upper triangular matrices. In this paper we want to study the action of on the set of monomial ideals of ( a field of characteristic zero) from a computational point of view. More specifically, we show that the stabilizer of a monomial ideal in is a purely combinatorial object and we give an algorithm for computing it. Then we characterize the subgroups of that are stabilizers of monomial ideals, we give an algorithm which finds if a given ideal is in the orbit of a monomial ideal under the action of and in the affirmative case, finds the matrices such that . We show that the entries of can be directly obtained from the coefficients of the generators of , so in particular no solutions of polynomial equations are required.