By Dietrich Kölzow

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**Example text**

P˜ ) is the matrix of = ni=1 εi ⊗ ei (resp. ˜ = n ∗ − ˜ ∈ zT +1 ⊗O ∗ if and only if i=1 ε˜ i ⊗ ei ) in the basis (e ⊗ e ). One has n v ⊗O (εi − ε˜ i ) ⊗ ei∗ ∗ T +1 i=1 which is clearly equivalent to P − P˜ ∈ zT +1 Mn (O). A similar argument holds for the second assertion of the lemma. 3. Any two matrices A, B ∈ Mn (K) satisfy v(AB) v(A) + v(B). Proof. Indeed one has n v(AB) = min v i,j Aik Bkj k=1 min min v(Aik ) + v(Bkj ) i,j k v(A) + v(B). 4. Let be a lattice in V , and (ε),(˜ε ) two bases of V .

E. P satisfies (Cq ). Call Tq the matrix obtained by the recollecting (q) of all blocks TI I . Clearly, the matrix Tq is a permutation matrix that commutes with A−p , . . , A−p+q−1 and the matrix A˜ = A[Tq ] satisfies A˜ |−p+q = B|−p+q . Hence the induction is finished. Note that the matrix T of the lemma is just the product T0 T1 . . Tp−1 and is therefore a permutation matrix. 14. Let A(z) = Mat(∇θ , (e)) = z1p A−p + zp−1 A−p+1 + · · · be the = L(e) is compatible with ∇. matrix of ∇θ in the basis (e).

Vs ) be a direct sum of V and is adapted to a lattice of V . Algorithmic computation of exponents for linear differential systems i) The lattice V which is adapted to V. is the largest sublattice of ii) V ∗ = (V1∗ , . . , Vs∗ ) is a direct sum of V ∗ and ∗ is adapted to V ∗ . iii) Let V ≺ W be a direct sum of V . If V 25 = then is adapted to V if and only if W = iv) Let W be a direct sum of V which is compatible with V. If V and W , then is adapted to V ∧ W . is adapted to both Proof. The lattice si=1 ( ∩Vi ) is adapted to the direct sum si=1 Vi according to its construction.