9 research outputs found
Key polynomials for simple extensions of valued fields
Let be a simple transcendental extension
of valued fields, where is equipped with a valuation of rank 1. That
is, we assume given a rank 1 valuation of and its extension to
. Let denote the valuation ring of . The purpose
of this paper is to present a refined version of MacLane's theory of key
polynomials, similar to those considered by M. Vaqui\'e, and reminiscent of
related objects studied by Abhyankar and Moh (approximate roots) and T.C. Kuo.
Namely, we associate to a countable well ordered set the are called {\bf key
polynomials}. Key polynomials which have no immediate predecessor are
called {\bf limit key polynomials}. Let .
We give an explicit description of the limit key polynomials (which may be
viewed as a generalization of the Artin--Schreier polynomials). We also give an
upper bound on the order type of the set of key polynomials. Namely, we show
that if then the set of key polynomials has
order type at most , while in the case
this order type is bounded above by , where stands
for the first infinite ordinal.Comment: arXiv admin note: substantial text overlap with arXiv:math/060519
Rank one discrete valuations of power series fields
In this paper we study the rank one discrete valuations of the field
whose center in k\lcor\X\rcor is the maximal ideal. In
sections 2 to 6 we give a construction of a system of parametric equations
describing such valuations. This amounts to finding a parameter and a field of
coefficients. We devote section 2 to finding an element of value 1, that is, a
parameter. The field of coefficients is the residue field of the valuation, and
it is given in section 5.
The constructions given in these sections are not effective in the general
case, because we need either to use the Zorn's lemma or to know explicitly a
section of the natural homomorphism R_v\to\d between the ring and
the residue field of the valuation .
However, as a consequence of this construction, in section 7, we prove that
k((\X)) can be embedded into a field L((\Y)), where is an algebraic
extension of and the {\em ``extended valuation'' is as close as possible to
the usual order function}
Extending a valuation centred in a local domain to the formal completion
International audienc
Resolution of Singularities: an Introduction.
International audienceThe problem of resolution of singularities and its solution in various contexts can be traced back to I. Newton and B. Riemann. This paper is an attempt to give a survey of the subject starting with Newton till the modern times, as well as to discuss some of the main open problems that remain to be solved. The main topics covered are the early days of resolution (fields of characteristic zero and dimension up to three), Zariski's approach via valuations, Hironaka's celebrated result in characteristic zero and all dimensions and its subsequent strenthenings and simplifications, existing resutls in positive characteristic (mostly up to dimension three), de Jong's approach via semi-stable reduction, Nash and higher Nash blowing up, as well as reduction of singuarities of vector fields and foliations. In many places, we have tried to summarize the main ideas of proofs of various results without getting too much into technical details