By F. Fauvet, C. Mitschi
9 examine papers from a March 2002 convention make clear a number of parts of combinatorics and dynamical platforms, with computing device algebra as an underlying and unifying subject matter. subject matters coated are abnormal connections, rank aid and summability of ideas of differential platforms, asymptotic habit of divergent sequence, integrability of Hamiltonian platforms, a number of zeta values, quasi polynomial formalism, Padè approximants with regards to analytic integrability, and hybrid platforms. The publication can be of curiosity to mathematicians and theoretical physicists. there isn't any topic index.
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Extra resources for From Combinatorics to Dynamical Systems: Journees de Calcul Formel, Strasbourg, March 22-23, 2002
Sample text
27, we have kn, ( L ) (n − 1)p. Therefore the announced result holds with T0 = (n − 1)p. 1) dz where A(z) ∈ Mn (K) is the matrix of the differential operator ∇θ in a basis (e) of V = K n . We only consider gauge transforms with respect to the derivative θ A[P ] = P −1 AP − P −1 θ P . 42 Eduardo Corel Since all the lattices introduced in the previous section are either maximal or minimal with respect to the original one, all gauge transforms P occurring in this section either satisfy v(P ) = 0 or v(P −1 ) = 0.
The following lemma summarizes basic properties of lattices and valuations (cf. [Cor2]). 1. Let = L(e) and M = L(ε) be two lattices in V . Let P denote the gauge matrix P(e),(ε) . Then the following hold. ,n v (εi ) = v(P ). ii) v (M) is the unique integer k such that z−k M ⊂ particular, one has v (z−v (M) M) = 0. and z−k−1 M ⊂ iii) For any integer k > 0, the lattice zk M is a strict sublattice of M. iv) M ⊂ if and only if P ∈ Mn (O) or, equivalently, if v(P ) 0. In 21 Algorithmic computation of exponents for linear differential systems v) M = if and only if P ∈ GLn (O), or equivalently, if P = P0 + P1 z + · · · with P0 ∈ GLn (C).
In this paper, we explicitly compute this normal form (and therefore the exponents) for a differential system whose determinant factors are not ramified. This can always be assumed after ramifying at an a priori known order (cf. [Le2]), but in practice, it remains a limitation. In Section 2 of this paper, we briefly recall the definitions of the more convenient framework of linear connections over vector spaces. In Section 3, we recall the results on which the characterization of exponents rely, and refer to [Cor2] for the proofs.