Algebraic power series and diagonals

AbstractWe extend some results of Christol and Furstenberg to the case of several variables: (1) A power series in several variables over the p-adic integers Zp is congruent mod ps to an algebraic power series if and only if its coefficients satisfy certain congruences mod ps. (2) Any algebraic power series in m variables over a field K can be written as the diagonal of a rational power series in 2m variables. We also give an elementary proof of a result of Deligne: The diagonal of an algebraic power series in several variables, over a field of nonzero characteristic is algebraic. Moreover we show that the diagonal of an algebraic power series over Zp is algebraic mod ps, for every s. We also obtain other related results