Jorge Martín-Morales

Gröbner bases and the number of Latin squares related to autotopisms of order ≤7

Latin squares can be seen as multiplication tables of quasigroups, which are, in general, non-commutative and non-associative algebraic structures. The number of Latin squares having a fixed isotopism in their autotopism group is at the moment an open problem. In this paper, we use Grobner bases to describe an algorithm that allows one to obtain the previous number. Specifically, this algorithm is implemented in Singular to obtain the number of Latin squares related to any autotopism of Latin squares of order up to 7.

Intersection Theory on Abelian-Quotient V-Surfaces and Q-Resolutions

In this paper we study the intersection theory on surfaces with abelian quotient singularities and we derive properties of quotients of weighted projective planes. We also use this theory to study weighted blow-ups in order to construct embedded

Cartier and Weil divisors on varieties with quotient singularities

\mathbf{Q}

Constructive D-Module Theory with Singular

-resolutions of plane curve singularities and abstract

Sudokus and Gröbner bases: not only a divertimento

\mathbf{Q}

Numerical adjunction formulas for weighted projective planes and lattice point counting

-resolutions of surfaces.

Coverings of Rational Ruled Normal Surfaces

It is well-known that the notions of Weil and Cartier Q-divisors coincide for V-manifolds. The main goal of this paper is to give a direct constructive proof of this result providing a procedure to express explicitly a Weil divisor as a rational Cartier divisor. The theory is illustrated on weighted projective spaces and weighted blow-ups.

Wirtinger curves, Artin groups, and hypocycloids

We overview numerous algorithms in computational D-module theory together with the theoretical background as well as the implementation in the computer algebra system Singular. We discuss new approaches to the computation of Bernstein operators, of logarithmic annihilator of a polynomial, of annihilators of rational functions as well as complex powers of polynomials. We analyze algorithms for local Bernstein–Sato polynomials and also algorithms, recovering any kind of Bernstein–Sato polynomial from partial knowledge of its roots. We address a novel way to compute the Bernstein–Sato polynomial for an affine variety algorithmically. All the carefully selected nontrivial examples, which we present, have been computed with our implementation. We also address such applications as the computation of a zeta-function for certain integrals and revealing the algebraic dependence between pairwise commuting elements.

Computing Volumes of Solids of Revolution with Double Integrals

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.

ON THE NUMBER OF IRREDUCIBLE COMPONENTS OF THE REPRESENTATION VARIETY OF A FAMILY OF ONE-RELATOR GROUPS

This paper gives an explicit formula for the Ehrhart quasi-polynomial of certain 2-dimensional polyhedra in terms of invariants of surface quotient singularities. Also, a formula for the dimension of the space of quasi-homogeneous polynomials of a given degree is derived. This admits an interpretation as a Numerical Adjunction Formula for singular curves on the weighted projective plane.