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A criterion for the similarity of length-two elements in a noncommutative PID. (English) Zbl 1236.16034

Summary: This paper presents a criterion for the similarity of length-two elements in a noncommutative principal ideal domain. The criterion enables the authors to develop an algorithm for determining whether \(B_1A_1\) and \(B_2A_2\) are similar, where \(A_1\), \(A_2\), \(B_1\), \(B_2\) are first-order differential (difference) operators. The main step in the algorithm is to find a rational solution of a parametric differential (difference) Risch’s equation, which has been well-studied in symbolic integration (summation).

MSC:

16U30 Divisibility, noncommutative UFDs
16Z05 Computational aspects of associative rings (general theory)
16S36 Ordinary and skew polynomial rings and semigroup rings
34A30 Linear ordinary differential equations and systems
68W30 Symbolic computation and algebraic computation
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References:

[1] O. Ore, Theory of non-commutative polynomials, Annals of Mathematics, 1933, 34(22): 480–508. · Zbl 0007.15101 · doi:10.2307/1968173
[2] M. van Hoeij, Rational solutions of the mixed differential equation and its application to factorization of differential operators, Proceedings ISSAC’96, ACM Press, 1996. · Zbl 0916.34015
[3] M. F. Singer, Testing reducibility of linear differential operators: A group theoretic perspective, Appl. Algebra Eng., Commun. Comput., 1996, 7(2): 77–104. · Zbl 0999.12007 · doi:10.1007/BF01191378
[4] N. Jacobson, Finite-Dimensional Division Algebras over Fields, Springer-Verlag, New York, 1996. · Zbl 0874.16002
[5] N. Jacobson, The Theory of Rings, American Math. Soc., New York, 1943. · Zbl 0060.07302
[6] A. J. Berrick and M. E. Keating, An Introduction to Rings and Modules: With K-theory in View, Cambridge University Press, Cambridge, 2000. · Zbl 0949.16001
[7] M. Bronstein and M. Petkovšek, An introduction to pseudo-linear algebra, Theoretical Computer Science, 1996, 157(1): 3–33. · Zbl 0868.34004 · doi:10.1016/0304-3975(95)00173-5
[8] M. Bronstein, Symbolic Integration I: Transcendental Functions, Springer-Verlag, New York, 2005. · Zbl 1059.12002
[9] C. Schneider, Solving parameterized linear difference equations in terms of indefinite nested sums and products, J. Differ. Equations Appl., 2005, 11(9): 799–821. · Zbl 1087.33011 · doi:10.1080/10236190500138262
[10] R. Risch, The problem of integration in finite terms, Trans. Amer. Math. Soc., 1969, 139: 167–189. · Zbl 0184.06702 · doi:10.1090/S0002-9947-1969-0237477-8
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