## 40 Citations

### Kronecker Function Rings of Transcendental Field Extensions

- Mathematics
- 2010

We consider the ring Kr(F/D), where D is a subring of a field F, that is the intersection of the trivial extensions to F(X) of the valuation rings of the Zariski–Riemann space consisting of all…

### Density of composite places in function fields and applications to real holomorphy rings

- MathematicsMathematische Nachrichten
- 2022

Given an algebraic function field F |K and a place ℘ on K, we prove that the places that are composite with extensions of ℘ to finite extensions of K lie dense in the space of all places of F , in a…

### Small irreducible components of arc spaces in positive characteristic

- MathematicsJournal of Pure and Applied Algebra
- 2022

### The constructible topology on spaces of valuation domains

- Mathematics
- 2012

We consider properties and applications of a compact, Hausdorff topology called the "ultrafilter topology" defined on an arbitrary spectral space and we observe that this topology coincides with the…

### On the geometry of Prüfer intersections of valuation rings

- Mathematics
- 2014

Let F be a field, let D be a subring of F and let Z be an irreducible subspace of the space of all valuation rings between D and F that have quotient field F. Then Z is a locally ringed space whose…

### Witt equivalence of function fields over global fields

- Mathematics
- 2015

In our work we investigate Witt equivalence of general function fields over global fields. It is proven that for any two such fields K and L the Witt equivalence induces a canonical bijection between…

### The structure of spaces of valuations and the local uniformization problem

- Mathematics
- 2013

The problem of resolution of singularities is a major problem in algebraic geometry. Local uniformization can be seen as its local version. For varieties over fields of characteristic zero, local…

### NOTES ON EXTREMAL AND TAME VALUED FIELDS

- MathematicsThe Journal of Symbolic Logic
- 2016

It is proved that in an extremal valued field of finite p-degree, the images of all additive polynomials have the optimal approximation property and can be used to improve the axiom system for the elementary theory of Laurent series fields over finite fields.

## References

SHOWING 1-10 OF 38 REFERENCES

### On local uniformization in arbitrary characteristic, I

- Mathematics
- 1999

We prove that every place of an algebraic function field F|K of arbitrary characteristic admits local uniformization in a finite extension F' of F. We show that F'|F can be chosen to be normal. If K…

### Embedding problems over large fields

- Mathematics
- 1996

In this paper we study Galois theoretic properties of a large class of fields, a class which includes all fields satisfying a universal local-global principle for the existence of rational points on…

### Value groups, residue fields, and bad places of rational function fields

- Mathematics
- 2004

We classify all possible extensions of a valuation from a ground field K to a rational function field in one or several variables over K. We determine which value groups and residue fields can…

### On places of algebraic function fields.

- Mathematics
- 1984

In this paper we study the space of all places of a function field in n variables. We denote by F/k an (algebraic) function field, i.e. F is a fmitely generated extension of k of transcendence degree…

### Local Uniformization on Algebraic Varieties

- Mathematics
- 1940

1. In [10] (p. 650) we have proved a uniformization theorem for zero-dimensional valuations on an algebraic surface, over an algebraically closed ground field K (of characteristic zero). In the…

### On valuation spectra

- Mathematics
- 1998

If K is an ordered field then every convex subring of K is a valuation ring of K. This easy but fundamental observation has made valuation theory a very natural and important tool in real algebraic…

### Prime ideal structure in commutative rings

- Mathematics
- 1969

0. Introduction. Let ' be the category of commutative rings with unit, and regard Spec (as in [1]) as a contravariant functor from ' to g$7 the category of topological spaces and continuous maps. The…

### Model theoretic methods in the theory of topological fields.

- Philosophy
- 1978

Suppose β is a base of τ. Then clearly the sentences (1) — (4) do not change their meaning when the variables U and F r nge over β instead of τ. The reason is that 3 F is used only in front of…

### Resolution of singularities : a research textbook in tribute to Oscar Zariski : based on the courses given at the working week in Obergurgl, Austria, September 7-14, 1997

- Mathematics
- 2000

Oscar Zariski 1899-1986.- Resolution of Singularities 1860-1999.- 1: Classes of the Working Week.- Alterations and resolution of singularities.- Reduction of singularities for differential…