Gorenstein curves and symmetry of the semigroup of values

LetO be the local ring of a irreducible algebroid curve and S its semigroup of values, Kunz in [7] proves thatO is a Gorenstein ring if and only if S is symmetrical. In this paper we give a generalization of this fact for the case of reduced curves with an arbitrary number of branches, d. For it we introduce a concept of symmetry for the semigroup of values S₊ ^d which generalizes the well known symmetry for d=1 (i.e. the irreducible case). This concept of symmetry is also closely related to the symmetry introduced by García in [4] (for the d=2 case) and the author in [3] (for arbitrary d) with the main goal of the explicit determination of S (in the case of plane curves).     

A generalized mean-value theorem

We give another proof of Seymour and Zaslavsky's theorem: For every familyf ₁,f ₂,...,f _n of continous functions defined on [0, 1], there exists a finite setF[0, 1] such that the average sum off _k overF coincides with the integral off _k for everyk=1, 2,...,n.