7,832 research outputs found

    Key polynomials for simple extensions of valued fields

    Full text link
    Let ι:KLK(x)\iota:K\hookrightarrow L\cong K(x) be a simple transcendental extension of valued fields, where KK is equipped with a valuation ν\nu of rank 1. That is, we assume given a rank 1 valuation ν\nu of KK and its extension ν\nu' to LL. Let (Rν,Mν,kν)(R_\nu,M_\nu,k_\nu) denote the valuation ring of ν\nu. 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 ι\iota a countable well ordered set Q={Qi}iΛK[x]; \mathbf{Q}=\{Q_i\}_{i\in\Lambda}\subset K[x]; the QiQ_i are called {\bf key polynomials}. Key polynomials QiQ_i which have no immediate predecessor are called {\bf limit key polynomials}. Let βi=ν(Qi)\beta_i=\nu'(Q_i). 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 char kν=0\operatorname{char}\ k_\nu=0 then the set of key polynomials has order type at most ω\omega, while in the case char kν=p>0\operatorname{char}\ k_\nu=p>0 this order type is bounded above by ω×ω\omega\times\omega, where ω\omega stands for the first infinite ordinal.Comment: arXiv admin note: substantial text overlap with arXiv:math/060519

    A generalization of Steinberg's cross-section

    Full text link
    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
    corecore