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\def\max{\mathop{\rm max}}
\def\min{\mathop{\rm min}}
\def\uple#1{(#1_1,\ldots,#1_n)}
\def\Dr{D^{(r)}}
%\def\D#1{D^{(#1)}}
\def\puple#1{(#1_1:\ldots:#1_n)}
\def\td{^t\!}
\def\deg{\mathop{\rm deg}}
\def\Gm{{\bf G}_m}
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\def\Z{{\bf Z}}
\def\zmp{\Z/p\Z}
\def\O#1{{{\cal O}_{#1}}}
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\def\mod{\ \hbox{mod}\,}
\def\Ext{\mathop{\hbox{Ext}}}
\centerline{\bf Differential operators and interpolation series in power
series fields}
\medskip
\centerline{\bf Jos\'e Felipe Voloch}
\bigskip

Let $k$ be a field and $K$ be the field of formal power series over $k$.
That is, the elements of $K$ are of the form $u = \sum_{n=n_0}^{\infty} a_nx^n$,
where $a_n \in k$ and $n_0$ is an arbitrary integer. If $a_{n_0} \neq 0$ we
put $v(u) = n_0$, then $v$ is a valuation on $K$ and $K$ is a local field,
i.e., it is complete with respect to this valuation. Let $U$ be an open subset
of $K$ and $f: U \to K$ a function. Besides the usual notion of continuity
there is the notion of differentiability for such functions $f$, namely, $f$
is differentiable in $a \in U$ if $\lim_{u \to a} (f(u) - f(a))/(u-a)$ exists.
A natural class of functions to consider is that of differential operators,
coming from differentiation with respect to the variable $x$. We can define
the Hasse derivations $\Dr, r \ge 0$ by:
$$\Dr(\sum a_nx^n) = \sum {n \choose r}a_nx^{n-r}.$$

\proclaim Theorem 1. The functions $\Dr: K \to K , r \ge 1$ are $k$-linear,
continuous and nowhere differentiable. (Differentiation is not differentiable!)

{\it Proof:} It is clear that $\Dr$ is $k$-linear and therefore it suffices
to check continuity and differentiability at $u = 0$. Plainly $v(\Dr(u)) \ge
v(u) - r$, so $\Dr$ is continuous (see also [Go], Prop. 13). 
Next, $\Dr$ is differentiable at $u = 0$ if
and only if $\lim_{u \to 0}\Dr(u)/u$ exists. However, the sequence $x^n$
converges to $0$ but $\Dr(x^n)/x^n = {n \choose r}x^{-r}$ does not converge.

Suppose now that $k$ is a finite field with $q$ elements. Then Wagner [W]
studied continuous linear functions $f: R \to K$, where $R = k[[x]]$. He 
obtained results analogous to classical results of Mahler [M] that gave 
interpolation series for continuous $p$-adic functions in terms of binomial
coefficients. To state Wagner's result we need to make a few definitions:

$$ \openup2pt  \displaylines{
\Psi_n (u) = \prod_{\textstyle {m \in k[x] \atop \deg m < n}}(u - m), n > 0,
\Psi_0 (u) = u, \cr
F_n = (x^{q^n} - x)(x^{q^{n-1}} - x)^q \cdots (x^q - x)^{q^{n-1}}, F_0 = 1, \cr
L_n = (x^{q^n} - x)(x^{q^{n-1}} - x) \cdots (x^q - x), L_0 = 1. \cr
}$$

Wagner then proved that every continuous linear function $f: R \to K$ can
be written as $ f = \sum_{n=0}^{\infty} A_n\Psi_n /F_n$,where $A_n \in K,
\lim_{n \to \infty}A_n = 0$ and moreover $f$ is
differentiable if and only if $\lim_{n \to \infty}A_n/L_n = 0$. The $A_n$ can
be obtained as follows. Define:

$$ \openup2pt  \displaylines{
\Delta_0f(u) = f(u) \cr
\Delta_{n+1}f(u) = \Delta_nf(xu) - x^{q^n}\Delta_nf(u). \cr
}$$

Wagner then shows that $A_n = \Delta_nf(1)$. Our next result computes the $A_n$
for the $\Dr$.

\proclaim Theorem 2.
For all $u \in R$ we have: $$\Dr(u) = \sum_{n=0}^{\infty}
A_{nr}{\Psi_n(u) \over {F_n}},$$ where $A_{n1} = (-1)^{n-1}L_{n-1}$ and, for 
$r > 1$, 
$$A_{nr} = (-1)^{n-1}L_{n-1}\sum_{0<i_1<\cdots<
i_{r-1}<n} {1 \over {(x-x^{q^{i_1}})\cdots(x-x^{q^{i_{r-1}}})}}.$$

{\it Proof:} We will show that $\Delta_n\Dr = \sum_{i=0}^{r-1}A_{n,r-i}D^{(i)}$
, for $n \ge 1$, by induction on $n$, and the result will follow from Wagner's 
results. Clearly, $\Delta_1\Dr = D^{(r-1)}$ so the above formula holds for
$n=1$. Assume the formula holds for $n$. From the recursive definition of 
$\Delta_{n+1}$ we get that $$\Delta_{n+1}\Dr = \sum_{i=0}^{r-1}(A_{n,r-i}
(x-x^{q^n}) + A_{n,r-i-1})D^{(i)} = \sum_{i=0}^{r-1}A_{n+1,r-i}D^{(i)}$$
and this completes the proof.

In particular we get the bizarre formula $du/dx = \sum_{n=0}^d (-1)^{n-1}L_{n-1}
\Psi_n(u)/F_n$ for $u \in k[x], \deg u \le d$.


Another class of continuous linear functions are $u \mapsto u\circ b$ for
$b \in xR$. These are differentiable when $b = x$ or $v(b) > 1$. The 
coefficients of their expansion in Wagner's basis are given by
$(b-x)\cdots(b-x^{q^{n-1}})$. The proof is left to the reader.

Finally we establish the following formula expanding the functions
$u^{q^i}$ in terms of the Hasse derivatives.

\proclaim Proposition. We have
$$u^{q^i}=\sum_{r=0}^\infty (x^{q^i}-x)^r \Dr u$$
for $u\in k[[x]]$.

{\it Proof:} We begin with the case $i=1$; i.e.,
$$u^q=\sum_{r=0}^\infty (x^q-x)^r\Dr u\,.$$
Note that both sides of this equation are $k$-linear,
so it suffices to check the formula for $u = x^m$ and in this case it is
straightforward. In this formula for $u^q$, one can replace $q$ by
$q^n$, for any $n$ by extending $k$ to its extension of degree $n$.
The proposition now follows.

{\bf Acknowledgments:} The author would like to thank D. Goss for suggestions
and encouragement.
The author would also like to acknowledge financial support by the NSF (grant no.
DMS-9301157) and by the NSA (grant MDA904-97-1-0037).
\bigskip
\centerline {\bf References}
\medskip
\noindent
[Go] D. Goss, {\it Separability, multi-valued operators, and zeroes of $L$-functions},
preprint, 1997.
\medskip
\noindent
[M] K. Mahler, {\it Introduction to $p$-adic numbers and their functions},
Cambridge Tracts in Mathematics {\bf 64}, Cambridge Univ. Press, Cambridge, 
1973.
\medskip
\noindent
[W] C. G. Wagner, {\it Linear operators in local fields of prime characteristic}
, Crelle {\bf 251}, 153-160.
\bigskip
\noindent
Dept. of Mathematics, Univ. of Texas, Austin, TX 78712, U.S.A.


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