  We provide a new proof of the multivariate version of Christol's theorem
about algebraic power series with coefficients in finite fields, as well as of
its extension to perfect ground fields of positive characteristic obtained
independently by Denef and Lipshitz, Sharif and Woodcok, and Harase. Our proof
is elementary, effective, and allows for much sharper estimates. We discuss
various applications of such estimates, in particular to a problem raised by
Deligne concerning the algebraicity degree of reductions modulo $p$ of
diagonals of multivariate algebraic power series with integer coefficients.
