By Brehm U., Kuhnel W.

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Let such a B be given. b) since [D−k , B˜ 0 ] = [T −1 B−k T , T −1 B0 T ] = 0 for all k = 1, . . , p. There is a gauge P such that B˜ [P ] = A. 11 shows that the gauge P satisfies conditions the gauge P applied to B. (Cp−1 ). Equation (p) is then given in B-blocks as (0) (0) −1 AI I = PI I (0) (0) B˜ I I PI I and (0) (0) −1 AI J = PI I (0) (0) (0) B˜ I J PJ J + PI I −1 (q− ) PI J (−p+ ) λI (−p+ ) − λJ Algorithmic computation of exponents for linear differential systems 49 (−p+ ) (−p+ ) − λJ = 0.

Since the non-integer part of the exponents is a meromorphic invariant of the attached D-module, one can give a definition of exponents modulo Z for regular systems. A proper notion of exponents, with a fixed integer part, at a regular singularity s has been defined by A. H. M. Levelt ([Le1]) as the maximal orders of growth e1s , . . 1) with respect to a suitable valuation. These exponents are usually impossible to compute without a previous complete computation of the solutions, except for the case of a matrix A with a simple pole at s.

Let be a lattice in V . For any 1 ki, ( · ) (resp. ki,. ( )) : L M −→ −→ i (M) , . . , zkn, (M) ). n, the map Z ki, (M) (resp. ki,M ( )) is a decreasing (resp. increasing) map, for the ordering of L by inclusion. Proof. Consider two lattices M ⊂ N . Fix a Smith basis (e) of for M and a Smith basis (ε) of for N . The gauge P˜ = z−K (N ) P(ε),(e) zK (M) is a gauge from N to M, hence P˜ ∈ Mn (O) and v(P˜ ) = mini,j v(Pij ) + kj, (N ) − ki, (M) 0. Since n, P ∈ GLn (O), there exists a permutation σ such that v(Pi,σ (i) ) = 0 for 1 i which implies kσ (i), (N)−ki, (M) 0.

### 15-vertex triangulations of an 8-manifold by Brehm U., Kuhnel W.

by Anthony

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