By Hendriks P.A.

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**Extra resources for Algebraic Aspects of Linear Differential and Difference Equations**

**Example text**

Let l be a K -linear map M ! K , extended as L-linear map L K M ! L such that l(v) = 0. From ii) we get lj (v) = 0 if j 62 Iv . Consider M (v) = (L C spanC (G:v))G. M (v) is #Iv -dimensional and lj (M (v)) = 0 if j 62 Iv . So M (v) = spanK feigi2Iv and l(M (v)) = 0 ) l(ei) = 0 for i 2 Iv . We conclude that M is linear Shidlovskii irreducible with respect to the K -base E = fe1 ; : : : ; eng, because lj (v) = 0 if j 62 Iv and l(ei) = 0 if i 2 Iv . 8 for Siegel normality. 9 Let M be a D-module and let f0g = M0 M1 Mr = M be a Jordan-Holder sequence.

Then the matrix T = (tij )i;j=1;:::;n 2 K nn is invertible and we have the following relation. A~ = TAT 1 + (T )T 1: The systems (A) and (A~) corresponding to the matrices A and A~ are dened to be equivalent if the above relation holds for a certain invertible matrix T 2 K nn. In that case if U 2 Lnn is a fundamental matrix of (A) then TU 2 Lnn is a fundamental matrix of the system (A~). Now it's obvious that the solution spaces V , V~ of the systems (A), (A~) are equivalent as representation spaces of the dierential Galois group DGal(L=K ).

After that we compute an appropriate equivalent system in standard form. 1 The Riccati equation To a second order linear dierence equation 2y + ay + by = 0, where a; b 2 K and b 6= 0 we can associate a rst order non-linear dierence equation (y) : u(u) + au + b = 0. This equation is called the Riccati equation. If u is a solution of the Riccati equation then the dierence operator 2 + a + b factors as ( ub )( u). A rational solution of the Riccati equation corresponds to a line in the solutionspace that is xed by the dierence Galois group.