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Introduction à la multisommabilité. (Introduction to multisummability). (French) Zbl 0722.34005
The paper is an informal introduction into the “multisummability theory” [J. Martinet and J.-P. Ramis, Computer algebra and differential equations, Colloq., Comput. Math. Appl., 117-214 (1988; Zbl 0722.12007)] and its applications in the theory of linear differential equations in the complex domain. From the fundamental formal matrix solution $$\hat Y(t)=\hat H(t)t^ L \exp Q(1/t)$$ of a linear differential system $$Y'=A(x)Y$$ where $$t=x^{1/p}$$ is a suitable ramification, one may obtain a “true solution” by summation of the formal matrix $$\hat H.$$ It is suggested that there exists a factorization of the matrix $$\hat H=\hat H_ 1\hat H_ 2...\hat H_ r$$ such that $$\hat H_ i$$ is $$k_ i$$- summable for every $$i=1,2,...,r$$. The theory is illustrated by several simple examples that are analyzed in some detail.

##### MSC:
 34M99 Ordinary differential equations in the complex domain 40G10 Abel, Borel and power series methods 12H05 Differential algebra