Document Type: Research Paper


Shiraz University


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


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