Bibliography

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

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: https://www.sciencedirect.com/science/article/abs/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.

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: https://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.

Bro

Manuel Bronstein. Poor Man's Integrator. URL: http://www-sop.inria.fr/cafe/Manuel.Bronstein/pmint/index.htm.

Bro05

Manuel Bronstein. Symbolic Integration I: Transcendental Functions. Springer–Verlag, New York, NY, USA, second edition, 2005. ISBN 3-540-21493-3.

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: https://dl.acm.org/doi/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: https://dl.acm.org/citation.cfm?doid=321371.321381, doi:10.1145/321371.321381.

CLOShea15

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.

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: https://www.sciencedirect.com/science/article/pii/S0747717183710515, doi:10.1006/jsco.1993.1051.

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.

GB73

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.

Gru96

Dominik Gruntz. On Computing Limits in a Symbolic Manipulation System. PhD thesis, Swiss Federal Institute of Technology, Zürich, Switzerland, 1996.

HNorsettW14

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.

JM09

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.

Kar81

Michael Karr. Summation in Finite Terms. Journal of the ACM, 28(2):305–350, 1981. doi:10.1145/322248.322255.

Knu85

Donald E. Knuth. The Art of Computer Programming: Seminumerical Algorithms. Addison–Wesley, Reading, MA, USA, second edition, 1985. ISBN 0-201-03822-6.

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: https://www.sciencedirect.com/science/article/pii/S0747717188800452, doi:10.1016/S0747-7171(88)80045-2.

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: https://www.sciencedirect.com/science/article/pii/S0747717183710539, 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: https://dl.acm.org/citation.cfm?doid=190347.190413, doi:10.1145/190347.190413.

MVV97

A. J. Menezes, O. P. C. Van, and S. A. Vanstone. Handbook of applied cryptography. CRC Press, Boca Raton, Florida, USA, 1997.

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.

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: https://dl.acm.org/citation.cfm?doid=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.

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://sites.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: https://dl.acm.org/citation.cfm?doid=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: https://dl.acm.org/citation.cfm?doid=258726.258784, doi:10.1145/258726.258784.

Sim16

G.F. Simmons. Differential Equations with Applications and Historical Notes, Third Edition. Textbooks in Mathematics. CRC Press, 2016. ISBN 9781498702621.

SW10

Yao Sun and Dingkang Wang. A new proof of the F5 algorithm. CoRR, 2010. URL: https://arxiv.org/abs/1004.0084, arXiv:1004.0084.

TP63

M. Tenenbaum and H. Pollard. Ordinary Differential Equations. Dover Publications, 1963.

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.

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: https://www.sciencedirect.com/science/article/pii/S0747717189800616, doi:10.1016/S0747-7171(89)80061-6.