Journal of Mathematical Sciences, Vol. 131, No. 5, 2005 UNIVARIATE ORE POLYNOMIAL RINGS IN COMPUTER ALGEBRA S. A. Abramov, H. Q. Le, and Z. Li UDC 512.55 Abstract. We present some algorithms related to rings of Ore polynomials (or, briefly, Ore rings) and describe a computer algebra library for basic operations in an arbitrary Ore ring. The library can be used as a basis for various algorithms in Ore rings, in particular, in differential, shift, and q-shift rings. CONTENTS 1. Introduction .............................................. 5885 2. Rings of Univariate Ore Polynomials ................................ 5886 3. Adjoint Operators .......................................... 5889 4. New Modular Techniques for gcd and lcm Computations ..................... 5892 5. The OreTools Package ........................................ 5897 6. Comparison .............................................. 5900 7. Availability .............................................. 5900 References ............................................... 5901 1. Introduction The theory of Ore rings gives us an opportunity to consider linear ordinary differential, difference, q-difference, and other operators from a uniform standpoint. This theory was proposed by Ore [24–26], who described, in particular, a uniform theory of the operator factorization, which generalizes the theory of Landau and Loewy for the differential case [17, 21, 22]. A way of interpreting abstract Ore polynomials as linear operators in a vector space was proposed by Jacobson [16]. The study of rings of Ore polynomials is attractive since it not only allows statements concerning operators of various kind to be proved in one stroke, but also allows one to design general-purpose algorithms and corresponding programs adjustable to a specific form of operators and equations. It is worth mentioning that the idea of using Ore rings in computer algebra was first employed by Bronstein and Petkovˇ sek in [8], where a general-purpose algorithm for factorization in an arbitrary Ore ring was described. In this paper, we describe a few (but far from all) computer algebra algorithms related to Ore rings. Section 2 provides an overview of rings of univariate Ore polynomials. The material of Sec. 3 on adjoint operators is presented in a more general form than in [3], while the contents of Sec. 4 on an efficient computation of greatest common divisors (gcd) and least common multiples (lcm) is presented for the first time. Section 5 gives an overview of the OreTools package, which provides facilities for working with univariate Ore polynomials in the Maple computer algebra system [23]. A comparison between this package and other related packages is done in Secs. 4 and 6. Information on the availability of the package is provided in Sec. 7. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 13, Algebra, 2004. 1072–3374/05/1315–5885 c  2005 Springer Science+Business Media, Inc. 5885