NON-ARCHIMEDEAN YOMDIN–GROMOV PARAMETRIZATIONS AND POINTS OF BOUNDED HEIGHT

We prove an analog of the Yomdin–Gromov lemma for $p$-adic definable sets and more broadly in a non-Archimedean definable context. This analog keeps track of piecewise approximation by Taylor polynomials, a nontrivial aspect in the totally disconnected case. We apply this result to bound the number of rational points of bounded height on the transcendental part of $p$-adic subanalytic sets, and to bound the dimension of the set of complex polynomials of bounded degree lying on an algebraic variety defined over $\mathbb{C}(\!(t)\!)$, in analogy to results by Pila and Wilkie, and by Bombieri and Pila, respectively. Along the way we prove, for definable functions in a general context of non-Archimedean geometry, that local Lipschitz continuity implies piecewise global Lipschitz continuity.