  We prove analogs of the Bezout and the Bernstein-Kushnirenko-Khovanskii
theorems for systems of algebraic differential conditions over differentially
closed fields. Namely, given a system of algebraic conditions on the first $l$
derivatives of an $n$-tuple of functions, which admits finitely many solutions,
we show that the number of solutions is bounded by an appropriate constant
(depending singly-exponentially on $n$ and $l$) times the volume of the Newton
polytope of the set of conditions. This improves a doubly-exponential estimate
due to Hrushovski and Pillay.
  We illustrate the application of our estimates in two diophantine contexts:
to counting transcendental lattice points on algebraic subvarieties of
semi-abelian varieties, following Hrushovski and Pillay; and to counting the
number of intersections between isogeny classes of elliptic curves and
algebraic varieties, following Freitag and Scanlon. In both cases we obtain
bounds which are singly-exponential (improving the known doubly-exponential
bounds) and which exhibit the natural asymptotic growth with respect to the
degrees of the equations involved.
