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.

翻译：我们给出了关于有限域系数代数幂级数的Christol定理的多元版本的新证明，并扩展至Denef与Lipshitz、Sharif与Woodcock以及Harase独立获得的关于正特征完美基域的推广。该证明方法初等且有效，能够导出显著更优的估计。我们讨论了这些估计的多种应用，特别涉及Deligne提出的关于整数系数多元代数幂级数对角线的模$p$约化代数性度问题。