\(E\)-algebraic functions over fields of positive characteristic -- an analogue of differentially algebraic functions
Let \(L\) be a differential field of characteristic zero. A function (or a power series) \(f\) is called differentially algebraic over \(L\) if it satisfies a differential equation of the form \(P(x,y,y',\dots,y^{(n)})=0\), where \(P\) is a nontrivial polynomial. This notion is not so significant over fields of characteristic \(p>0\), because \(f^{(p)}\equiv 0\). An analogue of the concept of a differentially algebraic power series is defined in the paper for a formal power series over a perfect field \(K\) of positive characteristic. It is shown that these series together with ordinary addition and multiplication of series form a field \(\Gamma_K\). The properties of this field are studied. It is shown that the field \(\Gamma_K\) is not closed under the Hadamard product operation (termwise multiplication of series).
differential fields of positive characteristic
scientific article; zbMATH DE number 1213797
scientific article