Integration in algebraically closed valued fields with sections

Yin, Yimu

doi:10.48550/arxiv.1204.5979

Cited by 5 publications

(29 citation statements)

References 9 publications

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“…From here on, our discussion will be of an increasingly formal nature. Many statements are exact copies of those in [19,20,22] and often the same proofs work, provided that the auxiliary results are replaced by the corresponding ones obtained above. For the reader's convenience, we will write down all the details.…”

Section: Generalized Euler Characteristicmentioning

confidence: 95%

“…Convention 5.1. We reiterate [22,Convention 2.32] here, with a different terminology, since this trivial-looking convention is actually quite crucial for understanding the discussion below, especially the parts that involve special bijections. For any set A, let c(A) = {(a, rv(a), t) : (a, t) ∈ A and a ∈ pr VF (A)}.…”

Section: Generalized Euler Characteristicmentioning

confidence: 99%

“…For a description of the ideas and the main results of the Hrushovski-Kazhdan style integration theory, we refer the reader to the original introduction in [10] and also the introductions in [20,22]. There is also a quite comprehensive introduction to the same materials in [11] and, more importantly, a specialized version that relates the Hrushovski-Kazhdan style integration to the geometry and topology of Milnor fibers over the complex field.…”

Section: Introductionmentioning

confidence: 99%

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Generalized Euler characteristic in power-bounded T-convex valued fields

Yin¹

2017

Compositio Math.

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We lay the groundwork in this first installment of a series of papers aimed at developing a theory of Hrushovski-Kazhdan style motivic integration for certain type of nonarchimedean o-minimal fields, namely power-bounded T -convex valued fields, and closely related structures. The main result of the present paper is a canonical homomorphism between the Grothendieck semirings of certain categories of definable sets that are associated with the VF-sort and the RV-sort of the language L T RV . Many aspects of this homomorphism can be described explicitly. Since these categories do not carry volume forms, the formal groupification of the said homomorphism is understood as a universal additive invariant or a generalized Euler characteristic. It admits, not just one, but two specializations to Z. The overall structure of the construction is modeled on that of the original Hrushovski-Kazhdan construction.

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Section: Generalized Euler Characteristicmentioning

confidence: 95%

Section: Generalized Euler Characteristicmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Generalized Euler characteristic in power-bounded T-convex valued fields

Yin¹

2017

Compositio Math.

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“…We introduce a syntactical complexity function µ, assigning a natural number to each quantifier-free formula (or a term) that indicates the highest nesting level of the function symbols exp, log : K −→ K, for instance, µ(exp(exp(X) log(X))) = 2; for this purpose there is no need to distinguish between exp and log. This is a standard syntactical device and we omit its definition here, see [9,Definition 2.16] for an analogue.…”

Section: Quantifier Eliminationmentioning

confidence: 99%

$T$-convex valued fields with tempered exponentiation

Yin¹

2020

Preprint

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We continue the effort of grokking the structure of power-bounded T -convex valued fields, whose theory is in general referred to as TCVF. In the present paper our focus is on certain expansion of it that is equipped with a tempered exponential function beyond the valuation ring. In order to construct such a tempered exponential function, the signed value group Γ is also converted into a model of T plus exponentiation and is in fact identified with (a section of) the residue field via the composition of a diagonal cross-section and an angular component map. In a sense, the resulting universal theory TKVF is a halfway point between power-bounded TCVF and exponential TCVF. This theory is reasonably well-behaved. In particular, we show that it admits quantifier elimination in a natural language, a notion of dimension, a generalized Euler characteristic, etc.

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“…With a reduced cross-section. We can add a reduced cross-section csn : Γ −→ RV to the language L RV , denote by L † RV the extension, and consider the corresponding integration theory; this has been worked out in [33]. We shall, however, only need a few facts about definable sets in RV in this setting.…”

Section: 41mentioning

confidence: 99%

Motivic integration and Milnor fiber

Fichou¹,

Yin²

2018

Preprint

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We put forward in this paper a uniform narrative that weaves together several variants of Hrushovski-Kazhdan style integral, and describe how it can facilitate the understanding of the Denef-Loeser motivic Milnor fiber and closely related objects. Our study focuses on the so-called "nonarchimedean Milnor fiber" that was introduced by Hrushovski and Loeser, and our thesis is that it is a richer embodiment of the underlying philosophy of the Milnor construction. The said narrative is first developed in the more natural complex environment, and is then extended to the real one via descent. In the process of doing so, we are able to provide more illuminating new proofs, free of resolution of singularities, of a few pivotal results in the literature, both complex and real. To begin with, the real motivic zeta function is shown to be rational, which yields the real motivic Milnor fiber; this is an analogue of the Hrushovski-Loeser construction. We also establish, in a much more intuitive manner, a new Thom-Sebastiani formula, which can be specialized to the one given by Guibert, Loeser, and Merle. Finally, applying T -convex integration after descent, matching the Euler Characteristics of the topological Milnor fiber and the motivic Milnor fiber becomes a matter of simple computation, which is not only free of resolution of singularities as in the Hrushovski-Loeser proof, but is also free of other sophisticated algebro-geometric machineries.

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Integration in algebraically closed valued fields with sections

Cited by 5 publications

References 9 publications

Generalized Euler characteristic in power-bounded T-convex valued fields

Generalized Euler characteristic in power-bounded T-convex valued fields

$T$-convex valued fields with tempered exponentiation

Motivic integration and Milnor fiber

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