Bibliography
The following is a non-comprehensive list of publications that were used as a theoretical foundation for implementing the Diofant.
Yu. A. Brychkov A. P. Prudnikov and O. I. Marichev. More Special Functions. Volume 3 of Integrals and Series. Gordon and Breach, 1990.
S.A. Abramov. On the summation of rational functions. USSR Computational Mathematics and Mathematical Physics, 11(4):324–330, 1971. URL: https://www.sciencedirect.com/science/article/abs/pii/0041555371900280, doi:10.1016/0041-5553(71)90028-0.
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.
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.
William Wells Adams and Philippe Loustaunau. An Introduction to Gröbner Bases. American Mathematical Society, Boston, MA, USA, July 1994. ISBN 0-821-83804-0.
Iyad A. Ajwa, Zhuojun Liu, and Paul S. Wang. Gröbner Bases Algorithm. Technical Report ICM-199502-00, ICM Technical Reports Series, 1995.
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: https://www.sciencedirect.com/science/article/pii/S0022314X96900499, doi:10.1006/jnth.1996.0049.
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.
Manuel Bronstein. Poor Man's Integrator. URL: http://www-sop.inria.fr/cafe/Manuel.Bronstein/pmint/index.htm.
Manuel Bronstein. Symbolic Integration I: Transcendental Functions. Springer–Verlag, New York, NY, USA, second edition, 2005. ISBN 3-540-21493-3.
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.
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.
W. S. Brown. The subresultant prs algorithm. ACM Transactions on Mathematical Software, 4(3):237–249, September 1978. URL: https://dl.acm.org/doi/10.1145/355791.355795, doi:10.1145/355791.355795.
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.
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.
George E. Collins. Subresultants and reduced polynomial remainder sequences. Journal of the ACM, 14(1):128–142, January 1967. URL: https://dl.acm.org/citation.cfm?doid=321371.321381, doi:10.1145/321371.321381.
David Cox, John Little, and Donald O'Shea. Ideals, Varieties and Algorithms. Undergraduate Texts in Mathematics. Springer–Verlag, New York, NY, USA, fourth edition, 2015. ISBN 978-3-319-16720-6.
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.
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: https://www.sciencedirect.com/science/article/pii/S0747717183710515, doi:10.1006/jsco.1993.1051.
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.
M.E. Goldstein and W.H. Braun. Advanced Methods for the Solution of Differential Equations. NASA (United States. National Aeronautics and Space Administration). Scientific and Technical Information Office, National Aeronautics and Space Administration, 1973.
Dominik Gruntz. On Computing Limits in a Symbolic Manipulation System. PhD thesis, Swiss Federal Institute of Technology, Zürich, Switzerland, 1996.
E. Hairer, S.P. Nørsett, and G. Wanner. Solving Ordinary Differential Equations I: Nonstiff Problems. Springer Series in Computational Mathematics. Springer–Verlag, 2014. ISBN 9783642052330.
Seyed Mohammad Mahdi Javadi and Michael Monagan. On Factorization of Multivariate Polynomials over Algebraic Number and Function Fields. In ISSAC '09: Proceedings of the 2009 International Symposium on Symbolic and Algebraic Computation, 199–206. New York, NY, USA, 2009. ACM Press. doi:10.1145/1576702.1576731.
Michael Karr. Summation in Finite Terms. Journal of the ACM, 28(2):305–350, 1981. doi:10.1145/322248.322255.
Donald E. Knuth. The Art of Computer Programming: Seminumerical Algorithms. Addison–Wesley, Reading, MA, USA, second edition, 1985. ISBN 0-201-03822-6.
W. Koepf. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, 1998.
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.
Heinz Kredel and Volker Weispfenning. Computing dimension and independent sets for polynomial ideals. Journal of Symbolic Computation, 6(2):231–247, 1988. URL: https://www.sciencedirect.com/science/article/pii/S0747717188800452, doi:10.1016/S0747-7171(88)80045-2.
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.
Yudell L. Luke. The Special Functions and Their Approximations. Volume 1. Academic Press, New York, NY, USA, 1969.
Yiu-Kwong Man. On computing closed forms for indefinite summations. Journal of Symbolic Computation, 16(4):355–376, October 1993. URL: https://www.sciencedirect.com/science/article/pii/S0747717183710539, doi:10.1006/jsco.1993.1053.
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: https://dl.acm.org/citation.cfm?doid=190347.190413, doi:10.1145/190347.190413.
A. J. Menezes, O. P. C. Van, and S. A. Vanstone. Handbook of applied cryptography. CRC Press, Boca Raton, Florida, USA, 1997.
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.
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: https://dl.acm.org/citation.cfm?doid=345542.345639, doi:10.1145/345542.345639.
B. Buchberger N.K. Bose and J.P. Guiver. Multidimensional Systems Theory and Applications. Springer, second edition, 2003.
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.
Marko Petkovšek, Herbert S. Wilf, and Doron Zeilberger. A \(=\) B. AK Peters, Ltd., Wellesley, MA, USA, 1997. URL: http://sites.math.rutgers.edu/~zeilberg/AeqB.pdf.
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: https://dl.acm.org/citation.cfm?doid=236869.237088, doi:10.1145/236869.237088.
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: https://dl.acm.org/citation.cfm?doid=258726.258784, doi:10.1145/258726.258784.
G.F. Simmons. Differential Equations with Applications and Historical Notes, Third Edition. Textbooks in Mathematics. CRC Press, 2016. ISBN 9781498702621.
Yao Sun and Dingkang Wang. A new proof of the F5 algorithm. CoRR, 2010. URL: https://arxiv.org/abs/1004.0084, arXiv:1004.0084.
M. Tenenbaum and H. Pollard. Ordinary Differential Equations. Dover Publications, 1963.
Barry M. Trager. Algebraic factoring and rational function integration. In SYMSAC '76: Proceedings of the third ACM Symposium on Symbolic and Algebraic Computation, 219–226. New York, NY, USA, 1976. ACM Press. doi:10.1145/800205.806338.
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.
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.
Kazuhiro Yokoyama, Masayuki Noro, and Taku Takeshima. Computing primitive elements of extension fields. Journal of Symbolic Computation, 8(6):553–580, 1989. URL: https://www.sciencedirect.com/science/article/pii/S0747717189800616, doi:10.1016/S0747-7171(89)80061-6.