Analytic roots of a rational function

Marie-Claude SARMANT

Let D = d ( 0,1^-) be the open unit disk of $K={\ \rm l\!\!\!C}_p$, let A(D) be the set of analytic functions in D and let $\widetilde D = d ( 0,1)$ be the unit disk ofK. If there exist $U \in A(D)$ with U(0) =0 and $P \in K[x]$ with P(0)=0, such that $1+ P(x) = ( 1+ U (x))^{p^2} \ \forall x \in D$, then $\Vert P \Vert$ is the absolute value of at least two coefficients of P. Consequently we can prove a conjecture asked by Escasssut, assuming an additive condition: Let $f \in H ( \widetilde D )$ be such that the differential equation y' = fy has a solution $g \in A (D)$ and that there exists an integer $N \not= 0$ such that g^N is an analytic element defined upon D. Then f is of the form : $f(x) = \d \sum^s_{j=1} \ {u_j \over x-c_j} + {dF\over dx} \quad$ with :

i)

$\forall j = 1,2, \ldots , s \qquad u_j \in { \rm l\!\!\!Q}\cap { \rm Z\!\!Z}_p$     and      $\vert c_j\vert> 1,$

ii)

$F \in A(D)$     and      $\Vert F\Vert \le p^{-{1\over p-1}}.$