Sharp derivatives for general sharply o-minimal structures

Determine whether the sharp derivatives condition holds for sharply o-minimal structures in general.

Background

The paper discusses an earlier proof of Wilkie’s conjecture that first establishes polylogarithmic point-counting bounds in a sharply o-minimal structure satisfying an additional condition called sharp derivatives. This condition controls the complexity of iterated derivatives of definable functions.

The authors note that sharp derivatives holds for the structure under consideration, but its validity for sharply o-minimal structures generally remains unresolved. Consequently, extending the point-counting framework to arbitrary sharply o-minimal structures would require resolving this condition or finding an alternative argument.