Geodesic
Hensel minimality II: Mixed characteristic and a diophantine application
Forum of Mathematics, Sigma, Tome 11 (2023)

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In this paper, together with the preceding Part I [10], we develop a framework for tame geometry on Henselian valued fields of characteristic zero, called Hensel minimality. It adds to [10] the treatment of the mixed characteristic case. Hensel minimality is inspired by o-minimality and its role in real geometry and diophantine applications. We develop geometric results and applications for Hensel minimal structures that were previously known only under stronger or less axiomatic assumptions, and which often have counterparts in o-minimal structures. We prove a Jacobian property, a strong form of Taylor approximations of definable functions, resplendency results and cell decomposition, all under Hensel minimality – more precisely, $1$-h-minimality. We obtain a diophantine application of counting rational points of bounded height on Hensel minimal curves. 
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     author = {Raf Cluckers and Immanuel Halupczok and Silvain Rideau-Kikuchi and Floris Vermeulen},
     title = {Hensel minimality {II:} {Mixed} characteristic and a diophantine application},
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Raf Cluckers; Immanuel Halupczok; Silvain Rideau-Kikuchi; Floris Vermeulen. Hensel minimality II: Mixed characteristic and a diophantine application. Forum of Mathematics, Sigma, Tome 11 (2023). doi: 10.1017/fms.2023.91

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