# Bibliography¶

The following is a non-comprehensive list of publications that were used as a theoretical foundation for implementing the Diofant.

 [Mat] Cyclotomic polynomial. http://mathworld.wolfram.com/CyclotomicPolynomial.html.
 [ABS02] B. Salvy A. Bostan, P. Flajolet and E. Schost. Fast computation with two algebraic numbers. Technical Report 4579, Institut National de Recherche en Informatique et en Automatique, 2002.
 [APPM90] Yu. A. Brychkov A. P. Prudnikov and O. I. Marichev. More Special Functions. Volume 3 of Integrals and Series. Gordon and Breach, 1990.
 [Abr71] S.A. Abramov. On the summation of rational functions. USSR Computational Mathematics and Mathematical Physics, 11(4):324–330, 1971. URL: http://www.sciencedirect.com/science/article/pii/0041555371900280, doi:10.1016/0041-5553(71)90028-0.
 [Abr95] Sergei A. Abramov. Rational Solutions of Linear Difference and q–difference Equations with Polynomial Coefficients. In ISSAC~‘95: Proceedings of the 1995 International Symposium on Symbolic and Algebraic Computation, 285–289. New York, NY, USA, 1995. ACM Press. doi:10.1145/220346.220383.
 [ABPetkovvsek95] Sergei A. Abramov, Manuel Bronstein, and Marko Petkovšek. On Polynomial Solutions of Linear Operator Equations. In ISSAC~‘95: Proceedings of the 1995 International Symposium on Symbolic and Algebraic Computation, 290–296. New York, NY, USA, 1995. ACM Press. doi:10.1145/220346.220384.
 [AL94] William Wells Adams and Philippe Loustaunau. An Introduction to Gröbner Bases. American Mathematical Society, Boston, MA, USA, July 1994. ISBN 0-821-83804-0.
 [ALW95] Iyad A. Ajwa, Zhuojun Liu, and Paul S. Wang. Gröbner Bases Algorithm. Technical Report ICM-199502-00, ICM Technical Reports Series, 1995.
 [Akr09] Alkiviadis Akritas. Linear and quadratic complexity bounds on the values of the positive roots of polynomials. J. UCS, 15:523–537, 01 2009.
 [ASV08] Alkiviadis Akritas, Adam Strzebonski, and Panagiotis Vigklas. Improving the performance of the continued fractions method using new bounds of positive roots. Nonlinear Analysis. Modelling and Control, 3:, 08 2008.
 [ARW96] Steven Arno, M.L. Robinson, and Ferell S. Wheeler. On denominators of algebraic numbers and integer polynomials. Journal of Number Theory, 57(2):292–302, 1996. URL: http://www.sciencedirect.com/science/article/pii/S0022314X96900499, doi:10.1006/jnth.1996.0049.
 [BW93] Thomas Becker and Volker Weispfenning. Gröbner Bases: A Computational Approach to Commutative Algebra. Volume 141 of Graduate Texts in Mathematics. Springer–Verlag, New York, NY, USA, 1993. ISBN 0-387-97971-9. In Cooperation with Heinz Kredel.
 [BO81] Michael Ben-Or. Probabilistic algorithms in finite fields. In Proceedings of the 22Nd Annual Symposium on Foundations of Computer Science, SFCS ‘81, 394–398. Washington, DC, USA, 1981. IEEE Computer Society. URL: https://doi.org/10.1109/SFCS.1981.37, doi:10.1109/SFCS.1981.37.
 [BS93] Manuel Bronstein and Bruno Salvy. Full Partial Fraction Decomposition of Rational Functions. In ISSAC~‘93: Proceedings of the 1993 International Symposium on Symbolic and Algebraic Computation, 157–160. New York, NY, USA, 1993. ACM Press. doi:10.1145/164081.164114.
 [Bro71] W. S. Brown. On Euclid’s Algorithm and the Computation of Polynomial Greatest Common Divisors. In SYMSAC~‘71: Proceedings of the second ACM Symposium on Symbolic and Algebraic Computation, 195–211. New York, NY, USA, 1971. ACM Press. doi:10.1145/800204.806288.
 [Bro78] W. S. Brown. The subresultant prs algorithm. ACM Transactions on Mathematical Software, 4(3):237–249, September 1978. URL: http://doi.acm.org/10.1145/355791.355795, doi:10.1145/355791.355795.
 [BT71] W. S. Brown and J. F. Traub. On Euclid’s Algorithm and the Theory of Subresultants. Journal of the ACM, 18(4):505–514, 1971. doi:10.1145/321662.321665.
 [Buc01] Bruno Buchberger. Gröbner Bases: A Short Introduction for Systems Theorists. In Computer Aided Systems Theory — EUROCAST 2001–Revised Papers, 1–19. London, UK, 2001. Springer–Verlag.
 [Col67] George E. Collins. Subresultants and reduced polynomial remainder sequences. Journal of the ACM, 14(1):128–142, January 1967. URL: http://doi.acm.org/10.1145/321371.321381, doi:10.1145/321371.321381.
 [CLOShea97] David Cox, John Little, and Donald O’Shea. Ideals, Varieties and Algorithms. Undergraduate Texts in Mathematics. Springer–Verlag, New York, NY, USA, second edition, 1997. ISBN 0-387-94680-2.
 [DST88] J. H. Davenport, Y. Siret, and E. Tournier. Computer algebra: systems and algorithms for algebraic computation. Academic Press, New York, NY, USA, 1988. ISBN 0-12-204230-1.
 [FaugereGLM93] J.C. Faugère, P. Gianni, D. Lazard, and T. Mora. Efficient computation of zero-dimensional gröbner bases by change of ordering. Journal of Symbolic Computation, 16(4):329–344, October 1993. URL: http://dx.doi.org/10.1006/jsco.1993.1051, doi:10.1006/jsco.1993.1051.
 [GCL92] Keith O. Geddes, Stephen R. Czapor, and George Labahn. Algorithms for Computer Algebra. Kluwer Academic Publishers, Norwell, MA, USA, 1992. ISBN 0-7923-9259-0.
 [GMN+91] Alessandro Giovini, Teo Mora, Gianfranco Niesi, Lorenzo Robbiano, and Carlo Traverso. “One sugar cube, please” or selection strategies in the Buchberger algorithm. In ISSAC~‘91: Proceedings of the 1991 International Symposium on Symbolic and Algebraic Computation, 49–54. New York, NY, USA, 1991. ACM Press. doi:10.1145/120694.120701.
 [Gru96] Dominik Gruntz. On Computing Limits in a Symbolic Manipulation System. PhD thesis, Swiss Federal Institute of Technology, Zürich, Switzerland, 1996.
 [KS98] Erich Kaltofen and Victor Shoup. Subquadratic-time factoring of polynomials over finite fields. Math. Comput., 67(223):1179–1197, July 1998. URL: http://dx.doi.org/10.1090/S0025-5718-98-00944-2, doi:10.1090/S0025-5718-98-00944-2.
 [Kar81] Michael Karr. Summation in Finite Terms. Journal of the ACM, 28(2):305–350, 1981. doi:10.1145/322248.322255.
 [Koe98] W. Koepf. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, 1998.
 [KL89] Dexter Kozen and Susan Landau. Polynomial Decomposition Algorithms. Journal of Symbolic Computation, 7(5):445–456, 1989. doi:10.1016/S0747-7171(89)80027-6.
 [KW88] Heinz Kredel and Volker Weispfenning. Computing dimension and independent sets for polynomial ideals. Journal of Symbolic Computation, 6(2):231–247, 1988. URL: http://www.sciencedirect.com/science/article/pii/S0747717188800452, doi:10.1016/S0747-7171(88)80045-2.
 [Lee13] M M-D Lee. Factorization of multivariate polynomials. PhD thesis, University of Kaiserslautern, 2013.
 [LF95] Hsin–Chao Liao and Richard J. Fateman. Evaluation of the heuristic polynomial GCD. In ISSAC~‘95: Proceedings of the 1995 International Symposium on Symbolic and Algebraic Computation, 240–247. New York, NY, USA, 1995. ACM Press. doi:10.1145/220346.220376.
 [Luk69] Yudell L. Luke. The Special Functions and Their Approximations. Volume 1. Academic Press, New York, NY, USA, 1969.
 [Man93] Yiu-Kwong Man. On computing closed forms for indefinite summations. Journal of Symbolic Computation, 16(4):355–376, October 1993. URL: http://dx.doi.org/10.1006/jsco.1993.1053, doi:10.1006/jsco.1993.1053.
 [MW94] Yiu-Kwong Man and Francis J. Wright. Fast polynomial dispersion computation and its application to indefinite summation. In Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC ‘94, 175–180. New York, NY, USA, 1994. ACM Press. URL: http://doi.acm.org/10.1145/190347.190413, doi:10.1145/190347.190413.
 [MvH04] Michael Monagan and Mark van Hoeij. Algorithms for Polynomial GCD Computation over Algebraic Function Fields. In ISSAC~‘04: Proceedings of the 2004 International Symposium on Symbolic and Algebraic Computation, 297–304. New York, NY, USA, 2004. ACM Press. doi:10.1145/1005285.1005328.
 [Mon93] Michael B. Monagan. In–place Arithmetic for Polinominals over $$Z_n$$. In DISCO ‘92: Proceedings of the International Symposium on Design and Implementation of Symbolic Computation Systems, 22–34. London, UK, 1993. Springer–Verlag.
 [MW00] Michael B. Monagan and Allan D. Wittkopf. On the design and implementation of brown’s algorithm over the integers and number fields. In Proceedings of the 2000 International Symposium on Symbolic and Algebraic Computation, ISSAC ‘00, 225–233. New York, NY, USA, 2000. ACM Press. URL: http://doi.acm.org/10.1145/345542.345639, doi:10.1145/345542.345639.
 [NKBG03] B. Buchberger N.K. Bose and J.P. Guiver. Multidimensional Systems Theory and Applications. Springer, second edition, 2003.
 [NZ91] I. Niven and H. S. Zuckerman. An Introduction to the Theory of Numbers. Wiley, 1991.
 [Petkovvsek92] Marko Petkovšek. Hypergeometric Solutions of Linear Recurrences with Polynomial Coefficients. Journal of Symbolic Computation, 14(2-3):243–264, 1992. doi:10.1016/0747-7171(92)90038-6.
 [PetkovvsekWZ97] Marko Petkovšek, Herbert S. Wilf, and Doron Zeilberger. A $$=$$ B. AK Peters, Ltd., Wellesley, MA, USA, 1997. URL: http://www.math.rutgers.edu/~zeilberg/AeqB.pdf.
 [Roa96] Kelly Roach. Hypergeometric function representations. In Proceedings of the 1996 International Symposium on Symbolic and Algebraic Computation, ISSAC ‘96, 301–308. New York, NY, USA, 1996. ACM Press. URL: http://doi.acm.org/10.1145/236869.237088, doi:10.1145/236869.237088.
 [Roa97] Kelly Roach. Meijer g function representations. In Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, ISSAC ‘97, 205–211. New York, NY, USA, 1997. ACM Press. URL: http://doi.acm.org/10.1145/258726.258784, doi:10.1145/258726.258784.
 [Sho91] Victor Shoup. A fast deterministic algorithm for factoring polynomials over finite fields of small characteristic. In Proceedings of the 1991 International Symposium on Symbolic and Algebraic Computation, ISSAC ‘91, 14–21. New York, NY, USA, 1991. ACM Press. URL: http://doi.acm.org/10.1145/120694.120697, doi:10.1145/120694.120697.
 [Sho95] Victor Shoup. A New Polynomial Factorization Algorithm and its Implementation. Journal of Symbolic Computation, 20(4):363–397, 1995. doi:10.1006/jsco.1995.1055.
 [SA05] Adam Strzebonski and Alkiviadis Akritas. A comparative study of two real root isolation methods. Nonlinear Analysis: Modelling and Control, 10:297–304, 01 2005.
 [SW10] Yao Sun and Dingkang Wang. A new proof of the F5 algorithm. CoRR, 2010. URL: http://arxiv.org/abs/1004.0084, arXiv:1004.0084.
 [VdH02] Joris Van der Hoeven. Relax, but don’t be too lazy. Journal of Symbolic Computation, 34(6):479–542, December 2002. URL: http://dx.doi.org/10.1006/jsco.2002.0562, doi:10.1006/jsco.2002.0562.
 [vHM02] Mark van Hoeij and Michael Monagan. A Modular GCD Algorithm over Number Fields Presented with Multiple Extensions. In ISSAC~‘02: Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation, 109–116. New York, NY, USA, 2002. ACM Press. doi:10.1145/780506.780520.
 [vzGG99] Joachim von zur Gathen and Jürgen Gerhard. Modern Computer Algebra. Cambridge University Press, New York, NY, USA, 1999. ISBN 0-521-64176-4.
 [vzGS92] Joachim von zur Gathen and Victor Shoup. Computing Frobenius Maps and Factoring Polynomials. In STOC~‘92: Proceedings of the twenty–fourth annual ACM Symposium on Theory of Computing, 97–105. New York, NY, USA, 1992. ACM Press. doi:10.1145/129712.129722.
 [Wan78] Paul S. Wang. An Improved Multivariate Polynomial Factoring Algorithm. Mathematics of Computation, 32(144):1215–1231, 1978.
 [Wan81] Paul S. Wang. A p–adic Algorithm for Univariate Partial Fractions. In SYMSAC~‘81: Proceedings of the fourth ACM Symposium on Symbolic and Algebraic Computation, 212–217. New York, NY, USA, 1981. ACM Press. doi:10.1145/800206.806398.
 [YNT89] Kazuhiro Yokoyama, Masayuki Noro, and Taku Takeshima. Computing primitive elements of extension fields. Journal of Symbolic Computation, 8(6):553–580, 1989. URL: http://www.sciencedirect.com/science/article/pii/S0747717189800616, doi:10.1016/S0747-7171(89)80061-6.