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Motivic Vitushkin invariants
- Part of
-
- Journal:
- Compositio Mathematica / Volume 161 / Issue 5 / May 2025
- Published online by Cambridge University Press:
- 20 August 2025, pp. 959-992
- Print publication:
- May 2025
-
- Article
- Export citation
-
Using motivic integration theory and the notion of riso-triviality, we introduce two new objects in the framework of definable nonarchimedean geometry: a convenient partial preorder
$\preccurlyeq$ on the set of constructible motivic functions, and an invariant 
$V_0$, nonarchimedean substitute for the number of connected components. We then give several applications based on 
$\preccurlyeq$ and 
$V_0$: we obtain the existence of nonarchimedean substitutes of real measure geometric invariants 
$V_i$, called the Vitushkin variations, and we establish the nonarchimedean counterpart of a real inequality involving 
$\preccurlyeq$, the metric entropy and our invariants 
$V_i$. We also prove the nonarchimedean Cauchy–Crofton formula for definable sets of dimension 
$d$, relating 
$V_0$ (and 
$V_d$) and the motivic measure in dimension 
$d$.
