Dr. Katz DEq Homework Solutions 55

# Dr. Katz DEq Homework Solutions 55 - (dr(p-1 k(p-1(p-k_l)k...

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Algebraic Solutions of Differential Equations 55 of two or more elements, dF(V, t2~./s(log D~)) contains ~sgn(i,A'~{i}) p-1 ~. [Ifl~ E (- If PP-l-k(f(i)) (3.5.4.13) i~a' k=l e;let=k, Ste)=A'-{i}~ E ] [I Lie (P)ev (a (v)) . ved'-{i} With no loss in generality, we may suppose A'= {1, 2 .... , b}, where b= ~(A'); then (3.5.4.14) sgn (i' A'A, {i}) =(- l) '+~. Let us introduce a final notation: (3.5.4.15) m=(mt . .... mb), a b-triple of integers satisfying mi>l, and Emi= p. A multi-index E occurring in (3.5.4.13) with I•l=k and S(~)=A'-{i} gives rise to such an m by putting mv=J~v, if v4=i (3.5.4.16) (p-k if v=i. In terms of m, the multinomial coefficient (- 1)k (IEI) is easily calculated:
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Unformatted text preview: (dr) (p-1), k! (p-1), (p-k) (_l)k I I _ k!(p-l-k)! fl!. ..fb! = ml!. ..mb! (3.5.4.17) -1 = 1-I (rod! mi modulo p. v Using (3.5.4.14) and (3.5.4.17), the congruence (3.5.4.13) may be rewritten -1 I-I (m~)! ~ (- 1)i§ miPm'-l(f(i)) (3.5.4.18) I-[ Lie(P)'v(a(v))edr(V,, OLs(l~ D~)). v*i The proof of (3.5.4.18), and hence of (3.5.0), will be concluded the following calculation. (3.5.5) Calculation. For any m as in (3.5.4.13), (-- 1) '+1 m i P"-~(f(i)) I-I Lie(P)m~(a(v)) i v*i (3.5.5.0) = d (P"~ -1 (f(1)) ~ ( - 1)i+1 rn, P"'- ~ (f(i)) 1-[ Lie (P)"~ (a (v))). i~kl v#l,i...
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