Dr. Katz DEq Homework Solutions 111

# Dr. Katz DEq Homework Solutions 111 - Algebraic Solutions...

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Algebraic Solutions of Differential Equations 111 (7.4.3.1) In order for the rank-one differential equation (K, V~,) on K to become trivial on a finite extension of K, it is necessary and sufficient that there exist an integer n > 1 and a function g~ K* such that n co = dg/g. (If n co = dg/g, then f= g-1/n is a non-zero algebraic solution of the equa- tion V~(f)=0. Conversely, iff is a non-zero algebraic solution, let n be the degree of K(f)/K. Then n co =-trace(d f/f)=dg/g, g= Norm(l/f).) (7.4.3.2) If (K, V~,) has p-curvature zero for almost all primes p (in the sense of (5.4.3.2)) (i. e., if co is ~ locally logarithmic mod p" for almost all primes p in the same sense (cf. (7.1.3.3)), then co has at worst first order poles, and rational residues (i. e., there is an integer n > 1 such that n co is a differential of the third kind on the complete non-singular model of K/k). (For by ([24], Theorem 13.0), the differential equation (K, V~,) has regular singular points, and rational exponents.) (7.4.4) Proposition.
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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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