Tesi Doctoral

Programa de doctorat Matemàtica Aplicada (UPC):

Títol:              On families of prime ideals with an unbounded minimal number of generators in a three-dimensional power series ring

Doctoranda:    LAURA GONZALEZ HERNANDEZ

Director:        FRANCESC D'ASSIS PLANAS VILANOVA

Resum

This thesis deals with the existence of families of prime ideals in the power series ring k[[x,y,z]] with an unbounded minimal number of generators.We begin by studying in-depth the related results of Moh on the area. We reprove and generalize a result of Moh which gives a lower bound on the minimal number of generators of an ideal in k[[x,y,z]]. In doing so, we demonstrate that the minimal number of generators of Moh’s prime P3 might decrease depending on the characteristic of the field k. This result contradicts a previous statement made by Sally and leaves as an open problem finding families of prime ideals in k[[x,y,z]] with an unbounded minimal number of generators, when the characteristic of k is different from zero. The main result of this thesis is the construction of a new family of prime ideals in k[[x,y,z]] with an unbounded minimal number of generators, explicitly described, up to constant coefficients, which improves all the former results. The construction and analysis of these families rely on the theory of numerical semigroups and the study of binomial matrices.We first study the numerical semigroup S spanned by three consecutive natural numbers, a,a+1,a+2. We define and characterize the set of elements whose factorizations have all the same length, ULF(S), We provide an explicit description of their factorization sets and a natural partition based on the length and the denumerant. Moreover, by using Apéry sets and Betti elements, we are able to extend some of these results to any general numerical semigroup G. These findings link the structural properties of S directly to the defining ideals of the semigroup rings k[t^a,t^b,t^c], providing a bridge between factorization theory and the minimal generating sets of the corresponding prime ideals.In addition to our particular study of the numerical semigroup S, we need to work with binomial matrices. We derive closed formulae for binomial determinants and calculate bases to left nullspaces of some special binomial matrices. Additionally, we provide an alternative proof for the positivity of binomial determinants, originally shown by Gessel and Viennot. Finally, we display our new family of prime ideals with unbounded minimal number of generators in k[[x,y,z]], where k is a field of characteristic zero. These primes are obtained as the kernel of a quasi-monomial algebra homomorphism. Up to constant coefficients, we give a description of their minimal generating polynomial sets. The advantage of our family with respect to some previous work is the explicit description of the minimal generating sets and the simplicity of the exponents of the monomial presentation.