scholarly journals Valuation Theoretic and Model Theoretic Aspects of Local Uniformization

2000 ◽  
pp. 381-456 ◽  
Author(s):  
Franz-Viktor Kuhlmann
Keyword(s):  
Local Uniformization

scholarly journals Every place admits local uniformization in a finite extension of the function field

2009 ◽  
Vol 221 (2) ◽  
pp. 428-453 ◽  
Author(s):  
Hagen Knaf ◽  
Franz-Viktor Kuhlmann
Keyword(s):  
Function Field ◽  
Finite Extension ◽  
Local Uniformization

An application of local uniformization to the theory of divisors

1951 ◽  
Vol 47 (2) ◽  
pp. 279-285
Author(s):  
D. G. Northcott
Keyword(s):  
Quotient Ring ◽  
Local Rings ◽  
Simple Point ◽  
Unique Factorization ◽  
The Third ◽  
Unique Factorization Domain ◽  
Irreducible Variety ◽  
Structure Theorems ◽  
Local Uniformization

If V is an irreducible variety and W is an irreducible simple subvariety of V, then one of the properties of the quotient ring of W in V is that it is a unique factorization domain. A proof of this theorem has been given by Zariski ((2), Theorem 5, p. 22), based on the structure theorems for complete local rings, and the fact that the local rings which arise geometrically are always analytically unramified. Here the theorem is deduced from certain properties of functions and their divisors which will be established by entirely different considerations. The terminology which will be employed is that proposed by A. Weil in his book(1), and we shall use, for instance, F-viii, Th. 3, Cor. 1, when referring to Corollary 1 of the third theorem in Chapter 8. Before proceeding to details it should be noted that Weil and Zariski differ in then-definitions, and that in particular the terms ‘variety’ and ‘simple point’ do not mean quite the same in the two theories. The effect of this is to make Zariski's result somewhat stronger than Theorem 3 of this paper.

scholarly journals Defect and local uniformization

2019 ◽  
Vol 113 (4) ◽  
pp. 4211-4226
Author(s):  
Steven Dale Cutkosky ◽  
Hussein Mourtada
Keyword(s):  
Local Uniformization

scholarly journals Local Uniformization of Abhyankar Valuations

2021 ◽  
Vol -1 (-1) ◽  
Author(s):  
Steven Dale Cutkosky
Keyword(s):  
Local Uniformization

scholarly journals Reduction of local uniformization to the case of rank one valuations for rings with zero divisors

2017 ◽  
Vol 66 (2) ◽  
pp. 277-293 ◽  
Author(s):  
Josnei Novacoski ◽  
Mark Spivakovsky
Keyword(s):  
Zero Divisors ◽  
Rank One ◽  
Local Uniformization

Local Uniformization on Algebraic Varieties

1940 ◽  
Vol 41 (4) ◽  
pp. 852 ◽  
Author(s):  
Oscar Zariski
Keyword(s):  
Algebraic Varieties ◽  
Local Uniformization

scholarly journals Truncated Local Uniformization of Formal Integrable Differential Forms

2021 ◽  
Vol 21 (1) ◽  
Author(s):  
F. Cano ◽  
M. Fernández-Duque
Keyword(s):  
Differential Forms ◽  
Newton Polygon ◽  
Codimension One ◽  
Inductive Procedure ◽  
Rank One ◽  
Local Uniformization ◽  
Rational Rank ◽  
Integrable Differential Forms

AbstractWe prove the existence of Local Uniformization for rational codimension one foliations along rational rank one valuations, in any ambient dimension. This result is consequence of the Truncated Local Uniformization of integrable formal differential 1-forms, that we also state and prove in the paper. Thanks to the truncated approach, we perform a classical inductive procedure, based both in the control of the Newton Polygon and in the possibility of avoiding accumulations of values, given by the existence of suitable Tschirnhausen transformations.

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