Density of algebraic points on Noetherian varieties

Abstract

Let $\Omega\subset{\mathbb R}^n$ be a relatively compact domain. A finite collection of real-valued functions on $\Omega$ is called a \emph{Noetherian chain} if the partial derivatives of each function are expressible as polynomials in the functions. A \emph{Noetherian function} is a polynomial combination of elements of a Noetherian chain. We introduce \emph{Noetherian parameters} (degrees, size of the coefficients) which measure the complexity of a Noetherian chain. Our main result is an explicit form of the Pila-Wilkie theorem for sets defined using Noetherian equalities and inequalities: for any $\epsilon>0$, the number of points of height $H$ in the transcendental part of the set is at most $C\cdot H^{\epsilon}$ where $C$ can be \emph{explicitly} estimated from the Noetherian parameters and $\epsilon$. We show that many functions of interest in arithmetic geometry fall within the Noetherian class, including elliptic and abelian functions, modular functions and universal covers of compact Riemann surfaces, Jacobi theta functions, periods of algebraic integrals, and the uniformizing map of the Siegel modular variety $\mathcal{A}_g$. We thus effectivize the (geometric side of) Pila-Zannier strategy for unlikely intersections in those instances that involve only compact domains.