… A third type of plane valuations which will be a useful tool for our purposes are the socalled
curve valuations: these plane valuations have Hamburger-Noether expansion with the shape …

… Then, for each a E S, the number of degree a generators of a minimal generating set of the
syzygy module Ni (i > 0) of gr,R as A[v]-module is 1 or 0. Moreover we can know whether this …

… Along this section o:groT → S denotes the weight function given by a plane valuation of type
B-II-a, C, D or E and by a minimal generating sequence {qi}i∈I of it. Recall that we have set …

… 3 and 41) and the sequences {Pi);=o ,,.., and {Piji=o ,,,., ,+I will be a minimal set of generators
of the semigroup of wj and of v, respectively (see [8, chap. 6 and 91). In the following, when …

Introduction THE subject-matter of this paper may be illustrated by an informal example. Let
us consider the plane curve branch F given in parametric form x= t\ y= t*+ t7. In the classical …

… of plane valuations is closely related to such algebraic curves, and the corresponding
valuations … As an example, we briefly show how to bound the minimum distance with the aid of …

… valuation µ est égale `a la valuation µι et il existe un polynôme φ = φι qui définit la valuation
µ comme valuation augmentée ou valuation … Nous disons dans ce cas que la valuation est …

… Lemma on which a major part of the work on valuations is based. Thus, a powerful tool,
already known to Newton in the discussion of plane curves, became available to algebra …

… of valuations corresponding to (reducible) plane curve … valuation of type II.2, the minimal
resolution will be defined as the minimal resolution of the corresponding divisorial valuation …

… plane valuations, introduces the class of plane valuations which we are interested in, plane
valuations … Since a valuation is determined by its minimum values at the maximal ideals mj of …