  The \emph{Noetherian class} is a wide class of functions defined in terms of
polynomial partial differential equations. It includes functions appearing
naturally in various branches of mathematics (exponential, elliptic, modular,
etc.). A conjecture by Khovanskii states that the \emph{local} geometry of sets
defined using Noetherian equations admits effective estimates analogous to the
effective \emph{global} bounds of algebraic geometry.
  We make a major step in the development of the theory of Noetherian functions
by providing an effective upper bound for the local number of isolated
solutions of a Noetherian system of equations depending on a parameter
$\epsilon$, which remains valid even when the system degenerates at
$\epsilon=0$. An estimate of this sort has played the key role in the
development of the theory of Pfaffian functions, and is expected to lead to
similar results in the Noetherian setting. We illustrate this by deducing from
our main result an effective form of the Lojasiewicz inequality for Noetherian
functions.
