By F. Fauvet, C. Mitschi

9 examine papers from a March 2002 convention make clear quite a few components of combinatorics and dynamical structures, with machine algebra as an underlying and unifying subject. issues lined are abnormal connections, rank relief and summability of recommendations of differential platforms, asymptotic habit of divergent sequence, integrability of Hamiltonian platforms, a number of zeta values, quasi polynomial formalism, Padè approximants on the topic of analytic integrability, and hybrid structures. The publication can be of curiosity to mathematicians and theoretical physicists. there isn't any topic index.

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**Extra info for From Combinatorics to Dynamical Systems: Journees de Calcul Formel, Strasbourg, March 22-23, 2002**

**Sample 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.