  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.
