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.