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By Hendriks P.A.

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53 4. If the Riccati equation has more than two solutions in K^  then the Riccati equation has in nitely many solutions and G is an algebraic subgroup of fc:Id j c 2 Q  g. Proof. 8. 1. 7 there exists an upper triangular matrix B 2 Gl(2; K ) and a matrix T = ( tt11 tt12 ) 2 Gl(2; K^ ) 21 22 0 1 such that B = (T )( b a )T 1: One can easily verify that t22 6= 0 and t ^ t 2 K satis es the Riccati equation. 21 22 2. If u is a solution of the Riccati equation then let T = ( 1 uu 11 ). We have B = (T )( 0b 1a )T 1 = ( u0  ) is an upper triangular matrix.

7 Let M be a D-module. Suppose dimK M = n  2. Further let G = DGal(M ). Then the following statements are equivalent: 1. M is simple and homogeneous algebraic Siegel normal. 2. 8 Let M be a D-module and let G = DGal(M ). Then the following statements are equivalent: 36 1. M is homogeneous algebraic Siegel normal. 2. M = ( L Mi ) L( L Nj ), where the Mi are are non-cogredient and noni=1 j =1 contragredient simple linear Siegel normal D-modules with dimK Mi  2 and the Nj are one dimensional D-modules satisfying the following condition: r s S k N1    S ks Ns ' Ntriv Ps implies either k ; : : : ; k = 0 or k 6= 0.

1 System (A) is called linear Shidlovskii irreducible if for any solution f = (f1; : : : ; fn)t of (A) and any pi 2 K the relation p1 f1 +    + pnfn = 0 implies for each i = 1; : : : ; n that either pi = 0 or fi = 0. In other words system (A) is linear Shidlovskii irreducible if the nonzero components of every solution f = (f1; : : : ; fn)t are linear independent over K . 2 System (A) is called homogeneous algebraic Shidlovskii irreducible if the nonzero components of any solution are homogeneous algebraic independent over K .

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Algebraic Aspects of Linear Differential and Difference Equations by Hendriks P.A.


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