# 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.

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.

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: 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.

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.

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.

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.