Abstract
We discuss the algorithms which, given a linear difference equation with rational function coefficients over a field k of characteristic 0, compute a polynomial U(x) ∈ k[x] (a universal denominator) such that the denominator of each of rational solutions (if exist) of the given equation divides U(x). We consider two types of such algorithms. One of them is based on constructing a set of irreducible polynomials that are candidates for divisors of denominators of rational solutions, and on finding a bound for the exponent of each of these candidates (the full factorization of polynomials is used). The second one is related to earlier algorithms for finding universal denominators, where the computation of gcd was used instead of the full factorization. The algorithms are applicable to scalar equations of arbitrary orders as well as to systems of first-order equations.
A complexity analysis and a time comparison of the algorithms implemented in Maple are presented.
Supported by ECONET grant 21315ZF.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abramov, S.: On the summation of rational functions. USSR Comput. Math. Phys. 11, 324–330 (1971); Transl. from Zh. vychisl. mat. mat. fyz. 11, 1071–1075 (1971)
Abramov, S.: Problems of computer algebra involved in the search for polynomial solutions of linear differential and difference equations. Moscow Univ. Comput. Math. Cybernet. 3, 63–68 (1989); Transl. from Vestn. MGU. Ser. 15. Vychisl. mat. i kibernet. 3, 53–60 (1989)
Abramov, S.: Rational solutions of linear difference and differential equations with polynomial coefficients. USSR Comput. Math. Phys. 29, 7–12 (1989); Transl. from Zh. vychisl. mat. mat. fyz. 29, 1611–1620 (1989)
Abramov, S.: Rational solutions of linear difference and q-difference equations with polynomial coefficients. In: ISSAC 1998 Proceedings, pp. 303–308 (1995)
Abramov, S.: Rational solutions of linear difference and q-difference equations with polynomial coefficients. Programming and Comput. Software 21, 273–278 (1995); Transl. from Programmirovanie 6, 3–11 (1995)
Abramov, S., Barkatou, M.: Rational solutions of first order linear difference systems. In: ISSAC 1998 Proceedings, pp. 124–131 (1998)
Abramov, S., Bronstein, M., Petkovšek, M.: On polynomial solutions of linear operator equations. In: ISSAC 1995 Proceedings, pp. 290–295 (1995)
Abramov, S., van Hoeij, M.: A method for the integration of solutions of Ore equations. In: ISSAC 1997 Proceedings, pp. 172–175 (1997)
Abramov, S., van Hoeij, M.: Integration of solutions of linear functional equations. Integral Transforms and Special Functions 8, 3–12 (1999)
Abramov, S., Ryabenko, A.: Indicial rational functions of linear ordinary differential equations with polynomial coefficients. Fundamental and Applied Mathematics 14(4), 15–34 (2008); Transl. from Fundamentalnaya i Prikladnaya Matematika 14(4), 15–34 (2008)
Barkatou, M.: Rational solutions of matrix difference equations: problem of equivalence and factorization. In: ISSAC 1999 Proceedings, pp. 277–282 (1999)
Von zur Gathen, J., Gerhard, J.: Modern Computer Algebra, 2nd edn. Cambridge University Press, Cambridge (2003)
Gerhard, J.: Modular Algorithms in Symbolic Summation and Symbolic Integration. LNCS, vol. 3218. Springer, Heidelberg (2004)
Gheffar, A., Abramov, S.: Valuations of rational solutions of linear difference equations at irreducible polynomials. Adv. in Appl. Maths (submitted 2010)
van Hoeij, M.: Rational solutions of linear difference equations. In: ISSAC 1998 Proceedings, pp. 120–123 (1998)
van Hoeij, M.: Factoring polynomials and the knapsack problem. J. Number Theory 95, 167–189 (2002)
van Hoeij, M., Levy, G.: Liouvillian solutions of irreducible second order linear difference equations. In: ISSAC 2010 Proc. (2010)
Khmelnov, D.E.: Search for polynomial solutions of linear functional systems by means of induced recurrences. Programming and Comput. Software 30, 61–67 (2004); Transl. from Programmirovanie 2, 8–16 (2004)
Knuth, D.E.: Big omicron and big omega and big theta. ACM SIGACT News 8(2), 18–23 (1976)
Man, Y.K., Wright, F.J.: Fast polynomial dispersion computation and its application to indefinite summation. In: ISSAC 1994 Proceedings, pp. 175–180 (1994)
Petkovšek, M.: Hypergeometric solutions of linear recurrences with polynomial coefficients. J. Symbolic Computation 14, 243–264 (1992)
Maple online help, http://www.maplesoft.com/support/help/
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Abramov, S.A., Gheffar, A., Khmelnov, D.E. (2010). Factorization of Polynomials and GCD Computations for Finding Universal Denominators. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2010. Lecture Notes in Computer Science, vol 6244. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15274-0_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-15274-0_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15273-3
Online ISBN: 978-3-642-15274-0
eBook Packages: Computer ScienceComputer Science (R0)