Submission date: 15 April 2021

Abstract:

We provide polynomial upper bounds for the minimal sizes of distal cell decompositions in several kinds of distal structures, particularly weakly o-minimal and P-minimal structures. The bound in general weakly o-minimal structures generalizes the vertical cell decomposition for semialgebraic sets, and the bounds for vector spaces in both o-minimal and p-adic cases are tight. We apply these bounds to Zarankiewicz's problem and sum-product bounds in distal structures.

Mathematics Subject Classification: 03C45 (Primary), 03C64, 05C35 (Secondary)

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Full text arXiv 2104.07769: pdf, ps.