Fractional ideals and integration with respect to the generalised Euler characteristic

Let $b$ be a fractional ideal of a one-dimensional Cohen-Macaulay local ring $O$ containing a perfect field $k$. This paper is devoted to the study some $O$-modules associated with $b$. In addition, different motivic Poincar\'e series are introduced by considering ideal filtrations associated with $b$; the corresponding functional equations of these Poincar\'e series are also described

The universal zeta function for curve singularities and its relation with global zeta functions

The purpose of this note is to give a brief overview on zeta functions of curve singularities and to provide some evidences on how these and global zeta functions associated to singular algebraic curves over perfect fields relate to each other.Comment: Survey on the "universal zeta function" defined for curve singularities by W. Z\'u\~niga and the author in their paper "Motivic zeta functions for curve singularities" [Nagoya Math. J. 198 (2010), 47-75]; a poster of it was presented in the "Workshop on Positivity and Valuations" held at the Centre de Recerca Matem\`atica, Barcelona, in 2016 February 22nd-26t

Lattice paths with given number of turns and semimodules over numerical semigroups

Let \Gamma= be a numerical semigroup. In this article we consider several relations between the so-called \Gamma-semimodules and lattice paths from (0,\alpha) to (\beta,0): we investigate isomorphism classes of \Gamma-semimodules as well as certain subsets of the set of gaps of \Gamma, and finally syzygies of \Gamma-semimodules. In particular we compute the number of \Gamma-semimodules which are isomorphic with their k-th syzygy for some k.Comment: 15 pages. Extended version of our previous submission "Lattice paths with given number of turns and numerical semigroups

Hilbert depth of graded modules over polynomial rings in two variables

In this article we mainly consider the positively Z-graded polynomial ring R=F[X,Y] over an arbitrary field F and Hilbert series of finitely generated graded R-modules. The central result is an arithmetic criterion for such a series to be the Hilbert series of some R-module of positive depth. In the generic case, that is, the degrees of X and Y being coprime, this criterion can be formulated in terms of the numerical semigroup generated by those degrees.Comment: 28 page

Fractional ideals and integration with respect to the generalised Euler characteristic

Let bb be a fractional ideal of a one-dimensional Cohen–Macaulay local ring containing a perfect field. This paper is devoted to the study of the motivic Poincaré series defined by different filtrations associated with bb in the form of Euler integrals with respect to the generalised Euler characteristic; in particular, the functional equations of the Poincaré series are also described.The author was partially supported by the Spanish Government “Ministerio de Economía y Competitividad” (MINECO), Grant MTM2012-36917-C03-03, and by the German Research Council (DFG), Grant GRK–1916