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84
N.M. Katz:
ordv (i~i fip ti) =mini(ordv(ff ti))
(6.0.6.1)
= min i (i + p ordvo (f~P)).
Let Kv (resp.
(KV)~o)
denote the completion of K (resp. K P) with
respect to the valuation v (resp. Vo). Then (6.0.6.0) gives
(6.0.6.2)
Kv ~ (KP)vo 0""
(KV)vo ~ K 
(KV),, o.
~
KP
primes
Since
(KP)vo = KW,
we have
(6.0.6.3)
K~= K  K~.
Let Soln(K) (resp. Soln(Ko)) denote the
K v
(resp. K p) vector space of
solutions of the differential equation (6.0.5.1) which lie in K (resp. in
Kv). Then
(6.0.6.4)
Soln
(Ko) ~
Soln (K)  K~.
Kp
This is because the differential operator
(6.0.6.5)
~
 ~ ai ~
: K* K
i=0
is KPlinear, and the differential operator
d
,
,1
d
i
is deduced from (6.0.6.5) by the (flat !) extension of scalars
K p ~ KW.
(6.0.6.7)
Because we can "clear denominators" by multiplying by
pth powers, the spaces Soln(K) and Soln(K~) are in fact
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This note was uploaded on 12/21/2011 for the course MAP 4341 taught by Professor Normankatz during the Fall '11 term at UNF.
 Fall '11
 NormanKatz

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