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Uniform Yomdin-Gromov parametrizations and points of bounded height in valued fields. (English) Zbl 1442.11098

Summary: We prove a uniform version of non-Archimedean Yomdin-Gromov parametrizations in a definable context with algebraic Skolem functions in the residue field. The parametrization result allows us to bound the number of \(\mathbb F_q[t]\)-points of bounded degrees of algebraic varieties, uniformly in the cardinality \(q\) of the finite field \(\mathbb F_q\) and the degree, generalizing work by Sedunova for fixed \(q\). We also deduce a uniform non-Archimedean Pila-Wilkie theorem [J. Pila and A. J. Wilkie, Duke Math. J. 133, No. 3, 591–616 (2006; Zbl 1217.11066)], generalizing work by Cluckers-Comte-Loeser [R. Cluckers et al., Geom. Funct. Anal. 20, No. 1, 68–87 (2010; Zbl 1220.12003)].

MSC:

11G50 Heights
03C98 Applications of model theory
11D88 \(p\)-adic and power series fields
11U09 Model theory (number-theoretic aspects)
14G05 Rational points
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References:

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