The Tree of Good Semigroups in {mathbb {N}}^2 and a Generalization of the Wilf Conjecture

Abstract

Good subsemigroups of {mathbb {N}}^d have been introduced as the most natural generalization of numerical ones. Although their definition arises by taking into account the properties of value semigroups of analytically unramified rings (for instance the local rings of an algebraic curve), not all good semigroups can be obtained as value semigroups, implying that they can be studied as pure combinatorial objects. In this work, we are going to introduce the definition of length and genus for good semigroups in {mathbb {N}}^d. For d=2, we show how to count all the local good semigroups with a fixed genus through the introduction of the tree of local good subsemigroups of {mathbb {N}}^2, generalizing the analogous concept introduced in the numerical case. Furthermore, we study the relationships between these elements and others previously defined in the case of good semigroups with two branches, as the type and the embedding dimension. Finally, we show that an analogue of Wilf’s conjecture fails for good semigroups in {mathbb {N}}^2.