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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
A generalization of Steinberg's cross-section
Let G be a semisimple group over an algebraically closed field. Steinberg has
associated to a Coxeter element w of minimal length r a subvariety V of G
isomorphic to an affine space of dimension r which meets the regular unipotent
class Y in exactly one point. In this paper this is generalized to the case
where w is replaced by any elliptic element in the Weyl group of minimal length
d in its conjugacy class, V is replaced by a subvariety V' of G isomorphic to
an affine space of dimension d and Y is replaced by a unipotent class Y' of
codimension d in such a way that the intersection of V' and Y' is finite.Comment: 21 page
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