E-Algebraic Functions over Fields of Positive Characteristic—An Analogue of Differentially Algebraic Functions (original) (raw)

. A function or a power series f is called differentially algebraic if it satisfies a Ž X Ž n. . differential equation of the form P x, y, y , . . . , y s 0, where P is a nontrivial polynomial. This notion is usually defined only over fields of characteristic zero and is not so significant over fields of characteristic p ) 0 as f Ž p. ' 0. For a formal power series over a perfect field K of 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 ⌫ with some natural properties. We also show that ⌫ is not closed under K K the Hadamard product operation. ᮊ 1998 Academic Press X Ž n.