Definable sets in equicharacteristic zero valued fields with analytic structure admit uniform T^r Yomdin-Gromov parametrizations with bound s times b_r^m.
Counting rational points on transcendental curves in valued fields
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abstract
We prove upper bounds on the number of rational points on transcendental curves in arbitrary $1$-h-minimal fields, similar to the Pila--Wilkie counting theorem in the o-minimal setting. These results extend results due to Cluckers--Comte--Loeser from $p$-adic fields to arbitrary valued fields of mixed characteristic. Our methods rely on parametrizations, where we avoid the usage of $r$-th power maps, combined with the determinant method.
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Strong stratifications and uniform Yomdin-Gromov parametrizations in valued fields with analytic structure
Definable sets in equicharacteristic zero valued fields with analytic structure admit uniform T^r Yomdin-Gromov parametrizations with bound s times b_r^m.