By Simons J., Sullivan D.

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Computing super-irreducible forms of systems of linear differential equations via moser-reduction: a new approach. In Proceedings of ISSAC ’07, pages 1–8, Waterloo, Canada, 2007. ACM Press. [13] M. A. Barkatou and E. Pfl¨ ugel. On the Moser– and super–reduction algorithms of systems of linear differential equations and their complexity. Submitted to JSC, 2007. [14] M. Bronstein and M. Petkovsek. An Introduction to Pseudo–Linear Algebra. Theoretical Computer Science, 157(1):3–33, 1996. [15] V. Dietrich.

The principle of the algorithm is given in Fig. 3, where evaluation and interpolation are performed by truncated-series products. In the case of n, n ∂ , the cor- (8) Likewise, the product of operators in K[X, X −1 ] θ reduces to some evaluation and interpolation tasks (in order to convert between operators and matrices) and to the main matrix-multiplication task (8), which is an instance of vC + dC + rC + 1, vA + dA + rC + 1, rC + 1 . The algorithm for multiplication in K[X] ∂ based on multiplication in K[X, X −1 ] θ is described in Fig.

Let now B1 and B2 be of bidegree (n, n) in K[X] ∂ . −1 Their conversion ` ´ in K[X, X ] θ can be performed using O n M(n) log n ops. and produces two operators C1 and C2 in K[X] θ , of bidegrees at most (2n, n) in (X, θ) such that B1 = X −n C1 (X, θ) and B2 = X −n C2 (X, θ). Using the commutation rule C2 (X, θ)X −n = X −n C2 (X, θ − n), we deduce the equalityPB2 B1 = X −2n C2 (X, θ − n)C1 (X, θ). Writing j C2 (X, θ) = n j=0 X cj (θ) shows that computing the coefficients of C2 (X, θ − n) amounts to n + 1 polynomial shifts in K[θ] in degree ` at most´ n.