Dr. Katz DEq Homework Solutions 82

# Dr. Katz DEq Homework Solutions 82 - 82 N.M Katz t hat...

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82 N.M. Katz: that whenever horizontal sections m w, one for each exponent-system W as above verify (6.0.3.1) ~ XW mw=O W then all mw= O. Let IWl=~ W~eZ, and put DW=[I i W~! in M ; this has a meaning because all W i__<p- 1. For any two exponent V, W as above, we have Dv(Xw)=o unless W~> V~ for all i (6.0.3.2) DV (x v) = 1 g ~ (~I(W~))X w-v if W~>V~ for all i. Dv ( x w) = Vi i Suppose that (6.0.3.1) holds, but that not all mw=O. Among exponent systems W having mw O, let V have a maximal weight IV]. Let's apply V(D v) to (6.0.3.1): (6.0.3.3) ~ V(Dv)(x w mw) = O. W Because all the mw are horizontal, V(Dv)(x w row)= Dv (Xw) row,
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