 By Hendriks P.A.

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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 cogredient or contragredient and deg trC(z)(f1 ; f10 ; g1; g10 ) = dimCDGal((A ; ; ); C(z)) = 3:  ~  Let f~i = zi +i fi and g~i = zi +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 = zi +i 1 zi +i .

If h = 0 everything is clear. Assume the statement holds for h = k 1. Let f0g = N0  N1      Nk = N be a Jordan-Holder 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 di erence equations of Kovacic's algorithm for second order linear di erential 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 di erence Galois theory which are needed for our purposes.