Dr. Katz DEq Homework Solutions 49

Dr. Katz DEq Homework Solutions 49 - dxi , if i~cz...

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Algebraic Solutions of Differential Equations 49 (3.5.0.2) we have (3.5.0.3) Then for any differential form e F (V, f2b/s (log D~)), Lie (P) k I(P- Q) Lie (Q)e (~ (z))- I(P p- Qp)(o~(z)) k+d=p-1 - ~ (I (P - Q) (~)) modulo dr(V, ~?s 1 (log D~)). Proof. Both sides of the asserted congruence (3.5.0.3) are p-linear in z, and exterior differentiation is linear over p-th powers, so it suffices to check the case in which z is a product of the one-forms (3.5.0.4) dx i i~ Xi dxi, iq~. We first introduce some auxilary notation:
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Unformatted text preview: dxi , if i~cz (3.5.0.5) z(i)= / xi dxi, if i~ct, [ dxi , if/ca (3.5.0.6) tr(i) = / xi x p-1 dx, if i~, [ P(x/) (3.5.0.7) f(i)=/ x~ ' if i~ (X p-I P(xi) if i~a, [ P(xi) if iea (3.5.0.8) g(i)=/ x~ ' tP(xi), if i~, [ PP(xi) , if iect (3.5.0.9) h(i)= / x, [x p-l Pv(xi), if i~e. (3.5.1) Lemma. z(i), 6(i), f (i), g(i), h(i) (3.5.1.1) (3.5.1.2) 4 Inventiones math., Vol. 18 The following relations hold among the "quantities" ~(z(i))=tr(i), Lie (P)(a (i)) = df(i),...
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