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Barkatou. Rational solutions of first order linear difference systems. In Proceedings of ISSAC ’98, pages 124–131, Rostock, Germany, 1998. ACM Press. [3] M. A. Barkatou. Contribution ` a l’´etude des ´equations diff´erentielles et de diff´erences dans le champ complexe. PhD thesis, INPG, 1989. [4] M. A. Barkatou. On the reduction of linear systems of difference equations. In Proceedings of ISSAC ’89, pages 1–6, Portland, Oregon, 1989. ACM Press. fr ABSTRACT number C of matrix multiplications with elements in K.

Thus´ using` Lemma´ 1(d), Step 2 is performed in O p2 M(n2 /p) ⊆ O p M(n2 ) ops. Note that Cu,v (X, Y ) has bidegree at most (2 n/p , 2n). To perform Step 3, each Cu,v (X p , θ)X v is first rewrit˜u,v (X p , θ) by computing 2 n/p + 1 shifts of ten as X v C polynomials of degree at most 2n. This can be done in ` ´ O pn M(n) log n ops. Finally, O(pn2 ) ops. are sufficient to Pp−1 u Pp−1 v ˜ p put C = u=0 X v=0 X Cu,v (X , θ) in canonical form. Summarizing, we have just proved: min(d−i,r) X (T )i X +i i ∂ .

Topics like multiplication of skew polynomials with unbalanced degrees and orders, or with sparse support, will be treated there. The case of rational (instead of polynomial) coefficients will also be considered. The methods of this article extend to multiplication of more general skew polynomials, in one or several variables, including for instance q-recurrences and partial differential operators. The constants in Table 1 are all somewhat pessimistic. Tighter bounds can be obtained by, on the one hand, relaxing the naive assumption (1), on the other hand, taking advantage of the special shapes (banded, trapezoidal, etc) of the various matrices.

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Algebra in a Local Topos With Applns to Ring Theory by F. Borceux, et al.,

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