Hilbert function, generalized Poincaré series and topology of plane valuations
Hilbert function, Hilbert series, Poincaré series and generalized Poincaré series attached to multi-index filtrations are objects which have been rather studied in a large series of interesting articles by the authors of this paper. This one keeps the same line and the authors prove that for filtrations given by what they call order functions (objects close to valuations that need not to satisfy \(\nu(fg)=\nu(f) + \nu(g)\)), Hilbert function, Hilbert series and generalized Poincaré series are equivalent in the sense that one of them determines the others. Notice that this is not the case of Poincaré series. The paper is also devoted to prove that the topological type of finite families of plane valuations (excluding curve valuations of rank 2) is determined by the mentioned generalized Poincaré series (and therefore by the Hilbert function).
scientific article; zbMATH DE number 6330893
scientific article