2016/05/15 by Gal Binyamini, Dmitry Novikov
Mathematics · #Advanced Topology and Set Theory #Algebra over a field #Arithmetic #Division (mathematics) #Functional Equations Stability Results #Holomorphic and Operator Theory #Mathematics #Pure mathematics #math.AG #math.LO #math.NT #msc:03C64 #msc:11G99 #msc:11U09
paper · pdf · doi:10.1112/s0010437x17007333
published as Compositio Math. 153 (2017) 2171-2194
arxiv created 2016/05/15 · openalex publication_date 2017/07/27 · arxiv updated 2019/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We present a complex analytic proof of the Pila–Wilkie theorem for subanalytic sets. In particular, we replace the use of Cr -smooth parametrizations by a variant of Weierstrass division. As a consequence we are able to apply the Bombieri–Pila determinant method directly to analytic families without limiting the order of smoothness by a Cr parametrization. This technique provides the key inductive step for our recent proof (in a closely related preprint) of the Wilkie conjecture for sets definable using restricted elementary functions. As an illustration of our approach we prove that the rational points of height H in a compact piece of a complex-analytic set of dimension k in ℂm are contained in O(1) complex-algebraic hypersurfaces of degree (log H)k/(m-k) . This is a complex-analytic analog of a recent result of Cluckers, Pila, and Wilkie for real subanalytic sets.