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arXiv · 2610.30762

Analytification and tropicalization via hyperfields and definable types

Pablo Cubides Kovacsics·Mariana Vicaría·Jinhe Ye
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Abstract

We study the analytification and tropicalization of algebraic varieties over valued fields via Jun's spaces of hyperfield points. Our main new objects are a family of power-tropical hyperfields $H_n$ and their inverse limit $H$, recording at each level the class of a leading term modulo $n$-th powers of the residue field. They yield an enhanced tropicalization of the associated space $X(H)$, strictly finer than the Berkovich one exactly when the multiplicative group of the residue field is non-divisible: it specialises to the Berkovich analytification when the residue field is algebraically closed, to the real analytification of Jell-Scheiderer-Yu when it is real closed, and to a new $p$-adic analytification when it is $p$-adically closed. Over henselian valued fields of equicharacteristic $0$ we then construct a model-theoretic avatar of $X(H)$: the space $\widehat{X}_T$ of definable types orthogonal to the value group, which sharpens the geometry in the spirit of Hrushovski and Loeser and recovers their space over algebraically closed valued fields. We show that $\widehat{X}_T$ is strictly pro-definable, satisfies a model-theoretic Payne theorem, and definable compactness in such spaces satisfies a Heine-Borel type characterization. As an application, we analytify $p$-adic semi-algebraic sets and prove a strong density theorem for their tropicalizations, the exact $p$-adic counterpart of the real one of Jell-Scheiderer-Yu.

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BibTeXRIS

Pablo Cubides Kovacsics, Mariana Vicaría, Jinhe Ye. 2026-10-22. Analytification and tropicalization via hyperfields and definable types. https://arxiv.org/abs/2610.30762

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