By Hendriks P.A.
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Additional info for Algebraic Aspects of Linear Differential and Difference Equations
Then 12 = 22 and either (1 1) + (2 2) 2 Z or (1 1) (2 2) 2 Z. 1 1 1 2 2 2 f f 1 g1 2 g2 Proof. Let U1 = f 0 g0 and U2 = f 0 g 0 be fundamental matrices of 1 1 2 2 the systems (A ; ; ) and (A ; ; ). Then 1 1 1 2 2 2 3 deg trC(z) (f1; f10 ; g1; g10 ; f2 ; f20 ; g2; g20 ) 4; because the systems (A ; ; ) and (A ; ; ) are cogredient or contragredient and deg trC(z)(f1 ; f10 ; g1; g10 ) = dimCDGal((A ; ; ); C(z)) = 3: ~ Let f~i = zi +i fi and g~i = zi +i gi, i = 1; 2 and let Vi= ff~i0 gg~~i0 , i = 1; 2, then i i i +i 1 0 z 0 Vi = Mi Ui, where Mi = 0 1 if i+i = 0 and else Mi = zi +i 1 zi +i .
If h = 0 everything is clear. Assume the statement holds for h = k 1. Let f0g = N0 N1 Nk = N be a Jordan-Holder sequence from f0g to N . As a consequence of the proof of 39 statement A) we have N1 N~ if multN N~ 1. Hence multS (N + N~ ) = multS (N + (N~ + N1 )) = multS (N=N1 + (N~ + N1)=N1) + SN = max(multS N=N1; multS (N~ + N1 )=N1) + SN = max(multS N; multS (N~ + N1 )) = max(multN N; multN N~ ); 1 1 1 1 1 where SN = 0 if S 6' N1 and SN = 1 if S ' N1 . The proof of multS (N \ N~ ) = min(multS N; multS N~ ) for each simple Dmodule S and any N; N~ 2 N is dual analogous.
This will be explained in the section which is devoted to examples. This algorithm can be considered as an analogue for second order linear dierence equations of Kovacic's algorithm for second order linear dierential equations (See [Kov86]). In [Pet92] an algorithm for nding rational solutions of the Riccati equation of arbitrary order is given. So the algorithm is not completely new. 3) seems to be new. 2 we summarize the results of [PS96] concerning dierence Galois theory which are needed for our purposes.