E-Algebraic Functions over Fields of Positive Characteristic—An Analogue of Differentially Algebraic Functions

A function (or a power series)fis called differentially algebraic if it satisfies a differential equation of the formP(x, y, y′,…,y(n)) = 0, wherePis a nontrivial polynomial. This notion is usually defined only over fields of characteristic zero and is not so significant over fields of characteristicp > 0 asf(p) ≡ 0. For a formal power series over a perfect fieldKof positive characteristic we shall define an analogue of the concept of a differentially algebraic power series. We shall show that these series together with ordinary addition and multiplication of series form a field ΓKwith some natural properties. We also show that ΓKis not closed under the Hadamard product operation.