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Algorithmic methods involving the rank of a differential operator. (Remarques algorithmiques liées au rang d’un opérateur différentiel linéaire.) (French)
Fauvet, F. (ed.) et al., From combinatorics to dynamical systems. Journées de calcul formel en l’honneur de Jean Thomann, Marseille, France, March 22‒23, 2002. Berlin: de Gruyter (ISBN 3-11-017875-3/pbk). IRMA Lect. Math. Theor. Phys. 3, 87-129 (2003).
According to {\it M. Loday-Richaud} [Expos. Math. 19, No. 3, 229‒250 (2001; Zbl 0990.34076)], “the reduction of the rank is a procedure which, to a linear differential system [\dots] with Poincaré rank $r$, associates a linear differential system [\dots] with Poincaré rank 1". This was introduced by {\it H. L. Turrittin} in [Duke. Math. J. 30, 271‒274 (1963; Zbl 0116.29005)]. A thorough theoretical study is expounded by {\it M. Loday-Richaud} [{loc. cit.}], where algorithmic aspects are also tackled. Among the questions thus raised is the following: given a series $\hat{f}(x) = \sum_{m \geq 0} u_{m} x^{m}$ that satisfies a given linear differential equation, find differential equations satisfied by the extracted series ${\hat{f}}^{j}(x) = \sum_{m \geq 0} u_{mr + j} x^{mr + j}$ [The formula given in p. 90 of the text reviewed here contains a small typo, the one in p. 92 is correct.] The authors discuss various algorithms that solve canonically this problem. After a first introductive section, in Section 2, the problem is solved for a scalar differential equation of order n, then for an equivalent linear system of rank n. In Section 3, one uses the duality of differential and difference systems provided by the Mellin transform and the same problems are solved for difference equations and systems. In each case, the underlying algebraic structures (Galois saturation and/or elimination) are emphasized. In Section 4, the various solutions are compared and found not to be equal: this is shown on the example of the equation of Ramis-Sibuya and explained. Also the complexity of the various algorithms is discussed. Last, a useful appendix collects some procedures to compute formal invariants of differential equations and systems. The paper also gives pointers to public MAPLE code for these algorithms.
Jacques Sauloy (Toulouse)