Abstract :
Let KK be a field of characteristic p 0p 0, K[[x]]K[[x]], the ring of formal power series over KK, K((x))K((x)), the quotient field of K[[x]]K[[x]], and K(x)K (x) the field of rational functions over KK. We shall give some characterizations of an algebraic function f∈K((x))f∈K ((x)) over KK. Let LL be a field of characteristic zero. The power series f∈L[[x]]f∈L[[x]] is called differentially algebraic, if it satisfies a differential equation of the form P(x,y,y′,...)=0P(x,y,y ′,...)=0, where PP is a non-trivial polynomial. This notion is defined over fields of characteristic zero and is not so significant over fields of characteristic p 0p 0, since f(p)=0f(p)=0. We shall define an analogue of the concept of a differentially algebraic power series over KK and we shall find some more related results.
