6533b7d6fe1ef96bd12658ca

RESEARCH PRODUCT

ÉQUATIONS DIFFÉRENTIELLES À COEFFICIENTS DANS DES CORPS DE SÉRIES GÉNÉRALISÉES.

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We express the connection between the support of some equations and those of generalized series solutions. On the one hand we prove that any real power series solution of a sub-analytic differential equation belong to a lattice (i.e. an additive sub semi-group of positive reals). On the other hand we consider the field Mr of series with well-ordered support included in the Hahn product Hr with finite rank r (i.e. the lexicographic product of r copies of the reals). We equip Mr with a "Hardy type" derivation and define some well-ordered sets T1, ..., Tr such that : for all equation F(y,...,y(n))=0 with F in Mr[[Y0,...,Yn]] and whose support Supp F is a well-ordered subset of Hr, and for all solution y0 in Mr with v(y0(i))> (0,...,0) for i=0,...,n, then the exponents of y0 belong to a positive well-ordered subset of Hr obtained from Supp F, T1, ..., Tr by a finite number of elementary transformations.