Integration in valued fields

By Ehud Hrushovski, David Kazhdan. 2005

139 citations recorded by the source corpus.

We develop a theory of integration over valued fields of residue characteristic zero. In particular, we obtain new and base-field independent foundations for integration over local fields of large residue characteristic, extending results of Denef, Loeser, and Cluckers. The method depends on an analysis of definable sets up to definable bijections. We obtain a precise description of the Grothendieck semigroup of such sets in terms of related groups over the residue field and value group. This yields new invariants of all definable bijections, as well as invariants of measure-preserving bijections.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/43e4ad22f01e627571341d30004248a43e554237

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Citations in the corpus

The motivic nearby fiber and degeneration of stable rationality (2017, background): “The following result is heavily inspired by Denef and Loeser’s motivic nearby fiber [11] and Hrushovski and Kazhdan’s theory of motivic integration [18]; the precise relations will be explained below.”
The Grothendieck ring of varieties and piecewise isomorphisms (2010, methodology): “Remark 7 After this paper is submitted, we found [18], Lemma 13.”
Constructible exponential functions, motivic Fourier transform and transfer principle (2005, unknown): “Note that in their recent paper [21] Hrushovski and Kazhdan have also considered integrals of exponentials.”
Constructible motivic functions and motivic integration (2004, background): “Also Hrushovski and Kazhdan [ 21 ] developed a general theory of integration for valued fields based on Robinson’s quantifier elimination for algebraically closed valued fields.”
On NIP and invariant measures (2007, background): “We consider the basic Cminimal theory consisting of a dense linear ordering (I,<) with greatest element∞, and another sort on which there are equivalence relationsEi indexed (uniformly) by i ∈ I and with Ej infinitely refining Ei if i < j (plus some other axioms, see [15]).”
Non-Archimedean Tame Topology and Stably Dominated Types (2010, background): “(See a fuller explanation in [19], introductory section on compactness and glueing.”