In 1984, Deligne proved that for any prime number $p$, the reduction modulo $p$ of the diagonal of a multivariate algebraic power series with integer coefficients is algebraic over the field of rational functions with coefficients in $\mathbb F_p$. Moreover, he conjectured that the algebraic degrees $d_p$ of these functions should grow at most polynomially in $p$. In this article, we provide a new and elementary proof of Deligne's theorem, which yields the first general polynomial bound on $d_p$ with an explicit and reasonable degree.

翻译：1984年，德利涅证明了对于任意素数 $p$，具有整数系数的多元代数幂级数的对角线模 $p$ 约化后，是系数在 $\mathbb F_p$ 上的有理函数域的代数元。此外，他猜想这些函数的代数次数 $d_p$ 的增长至多是 $p$ 的多项式。本文中，我们给出了德利涅定理的一个全新且初等的证明，该证明首次为 $d_p$ 提供了具有显式且合理次数的多项式一般界。