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

Part of: Birational geometry Topological fields Model theory

Published online by Cambridge University Press:  14 September 2017

Yimu Yin
Yimu Yin*
Affiliation:
Department of Philosophy, Sun Yat-Sen University, 135 Xingang Road West, Guangzhou 510275, China email [email protected]
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Abstract

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 types 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 $\text{VF}$-sort and the $\text{RV}$-sort of the language ${\mathcal{L}}_{T\text{RV}}$. 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 $\unicode[STIX]{x2124}$. The overall structure of the construction is modeled on that of the original Hrushovski–Kazhdan construction.

Keywords

motivic integrationEuler characteristico-minimal valued fieldT-convexity

MSC classification

Primary: 12J25: Non-Archimedean valued fields 03C64: Model theory of ordered structures; o-minimality 14E18: Arcs and motivic integration 03C98: Applications of model theory
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 12 , December 2017 , pp. 2591 - 2642
Copyright
© The Author 2017 

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