| Title |
Valuation Theory in Interaction ([EBook] /) / Antonio Campillo, Franz-Viktor Kuhlmann, Bernard Teissier |
| Added Personal Name |
Campillo, Antonio |
| Kuhlmann, Franz-Viktor |
| Teissier, Bernard |
| Publication |
Zuerich, Switzerland : : European Mathematical Society Publishing House, , 2014 |
| Physical Details |
1 online resource (670 pages) |
| Series |
EMS Series of Congress Reports (ECR) 2523-515X |
| ISBN |
9783037196496 |
| Summary Note |
Having its classical roots, since more than a century, in algebraic number theory, algebraic geometry and the theory of ordered fields and groups, valuation theory has seen an amazing expansion into many other areas in recent decades. Moreover, having been dormant for a while in algebraic geometry, it has now been reintroduced as a tool to attack the open problem of resolution of singularities in positive characteristic and to analyse the structure of singularities. Driven by this topic, and by its many new applications in other areas, also the research in valuation theory itself has been intensified, with a particular emphasis on the deep open problems in positive characteristic. As important examples for the expansion of valuation theory, it has become extremely useful in the theory of complex dynamical systems, and in the study of non-oscillating trajectories of real analytic vector fields in three dimensions. Analogues of the Riemann-Zariski valuation spaces have been found to be the proper framework for questions of intersection theory in algebraic geometry and in the analysis of singularities of complex plurisubharmonic functions. In a different direction, the relation between Berkovich geometry, tropical geometry and valuation spaces, on the one hand, and the geometry of arc spaces and valuation spaces, on the other, have begun to deepen and clarify. Ever since its beginnings, valuation theory and Galois theory have grown closely together and influenced each other. Arguably, studying and understanding the extensions of valuations in algebraic field extensions is one of the most important questions in valuation theory, whereas using valuation theory is one of he most important tools in studying Galois extensions of fields, as well as constructing field extensions with given properties. The well established topic of the model theory of valued fields is also being transformed, in particular through the study of valued fields with functions and operators, a...: |
| Contents note |
A study of irreducible polynomials over henselian valued fields via distinguished pairs /: On fields of totally $\mathfrak{S}$-adic numbers. With an appendix by Florian Pop /: Infinite towers of Artin-Schreier defect extensions of rational function fields /: A refinement of Izumi's Theorem /: Multivariable Hodge theoretical invariants of germs of plane curves. II /: Existence des diviseurs dicritiques, d’après S.S. Abhyankar /: Invariants of the graded algebras associated to divisorial valuations dominating a rational surface singularity /: An introduction to $C$-minimal structures and their cell decomposition theorem /: Valuation semigroups of Noetherian local domains /: Additive polynomials over perfect fields /: On $\mathbb{R}$-places and related topics /: Extending valuations to formal completions /: Extending real valuations to skew polynomial rings /: Stratifications in valued fields /: Imaginaries and definable types in algebraically closed valued fields /: Defects of algebraic function fields, completion defects and defect quotients /: On generalized series fields and exponential-logarithmic series fields with derivations /: Jet schemes of rational double point singularities /: Valuations centered at a two-dimensional regular local domain: infima and topologies /: Reduction of local uniformization to the rank one case /: Little survey on large fields - old & new /: Quasi-valuations -- topology and the weak approximation theorem /: Overweight deformations of affine toric varieties and local uniformization /: Detecting valuations using small Galois groups /: Truncation in Hahn fields /: The ergodicity of 1-Lipschitz transformations on 2-adic spheres /: Ramification of higher local fields approaches and questions /: |
| Mode of acces to digital resource |
Mode of access: World Wide Web. System requirements: Internet Explorer 6.0 (or higher) or Firefox 2.0 (or higher). Available as searchable text in PDF format. |
| System details note |
Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users). |
| Internet Site |
https://doi.org/10.4171/149 |
| See Also |
https://www.ems-ph.org/img/books/campillo_mini.gif |
| Links to Related Works |
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