  We prove a quantitative version of a result of Furstenberg and Deligne
stating that the the diagonal of a multivariate algebraic power series with
coefficients in a field of positive characteristic is algebraic. As a
consequence, we obtain that for every prime $p$ the reduction modulo $p$ of the
diagonal of a multivariate algebraic power series $f$ with integer coefficients
is an algebraic power series of degree at most $p^{A}$ and height at most
$A^2p^{A+1}$, where $A$ is an effective constant that only depends on the
number of variables, the degree of $f$ and the height of $f$. This answers a
question raised by Deligne.
