Dr. Katz DEq Homework Solutions 84

Dr Katz DEq - 84 N M Katz ordv(i~i fip ti =mini(ordv(ff ti( = min i(i p ordvo(f~P L et Kv(resp(KV)~o d enote the completion of K(resp K P

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84 N.M. Katz: ordv (i~i fip ti) =mini(ordv(ff ti)) ( = 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 ( gives ( Kv ~- (KP)vo 0"" (KV)vo ~- K | (KV),, o. ~ KP primes Since (KP)vo = KW, we have ( K~= K | K~. Let Soln(K) (resp. Soln(Ko)) denote the K v (resp. K p) vector space of solutions of the differential equation ( which lie in K (resp. in Kv). Then ( Soln (Ko) ~ Soln (K) | K~. Kp This is because the differential operator ( -~- - ~ ai -~- : K-* K i=0 is KP-linear, and the differential operator d , ,-1 d i is deduced from ( by the (flat !) extension of scalars K p ~ KW. ( Because we can "clear denominators" by multiplying by p-th 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.

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