Abstract:
Numerical semigroups are additive submonoids of (N,+), with finite complement in N. Despite their simple structure, there are many interesting problems regarding numerical semigroups, often arising from their connection with other research subjects in mathematics. In this talk I will present a longstanding conjecture known as Wilf’s conjecture, that is an inequality involving embedding dimension and multiplicity of a numerical semigroup. This conjecture has been proved for some classes of numerical semigroups, but no general strategy has been found to solve it. Successively, I will show a similar, but weaker, inequality, recently proved by A. Moscariello and myself. This inequality, like also Wilf's conjecture, can be re-interpreted for one-dimensional local rings and also in higher dimension, provided that the conductor of the ring in its integral closure in the total ring of fractions is m-primary. The new results presented in this talk are contained in two joint works with A. Moscariello.