Analytification and tropicalization via hyperfields and definable types
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 Hn 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 X̂T 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 X̂T 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.
Related
BibTeX
@misc{kovacsics2026analytification,
title = {{Analytification and tropicalization via hyperfields and definable types}},
author = {Pablo Cubides Kovacsics and Mariana Vicaría and Jinhe Ye},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30762},
eprint = {2610.30762},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30762},
}
RIS
TY - UNPB
TI - Analytification and tropicalization via hyperfields and definable types
AU - Pablo Cubides Kovacsics
AU - Mariana Vicaría
AU - Jinhe Ye
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.30762
UR - https://arxiv.org/abs/2610.30762
AB - 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 X̂_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 X̂_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.
ER -