Commutative Algebra

Computational Methods

13-04

  1. I. Abdeljaouad-Tej, S. Orange, G. Renault, and A. Valibouze, Computation of the decomposition group of a triangular ideal, Appl. Algebra Engrg. Comm. Comput. 15 (2004), no. 3-4, 279–294.[MR]
  2. Fatima Abu Salem, Shuhong Gao, and Alan G. B. Lauder, Factoring polynomials via polytopes, ISSAC 2004, ACM, New York, 2004, pp. 4–11.[MR]
  3. Gwénolé Ars, Jean-Charles Faugère, Hideki Imai, Mitsuru Kawazoe, and Makoto Sugita, Comparison between XL and Gröbner basis algorithms, Advances in Cryptology—Asiacrypt 2004, Lecture Notes in Comput. Sci., vol. 3329, Springer, Berlin, 2004, pp. 338–353.[MR]
  4. Philippe Aubry and Marc Moreno Maza, Triangular sets for solving polynomial systems: A comparative implementation of four methods, J. Symbolic Comput. 28 (1999), no. 1-2, 125–154.[MR]
  5. Mohamed Ayad and Peter Fleischmann, On the decomposition of rational functions, J. Symbolic Comput. 43 (2008), no. 4, 259–274.[MR]
  6. Bernd Bank, Marc Giusti, Joos Heintz, Mohab Safey El Din, and Eric Schost, On the geometry of polar varieties, Appl. Algebra Engrg. Comm. Comput. 21 (2010), no. 1, 33–83.[MR/doi]
  7. Aurélie Bauer and Antoine Joux, Toward a rigorous variation of Coppersmith's algorithm on three variables, Advances in cryptology—EUROCRYPT 2007, Lecture Notes in Comput. Sci., vol. 4515, Springer, Berlin, 2007, pp. 361–378.[MR]
  8. Karim. Belabas, Mark van Hoeij, J. Klüners, and Allan Steel, Factoring polynomials over global fields, Journal de Théorie des Nombres de Bordeaux (2009), no. 21, 15–39.
  9. Thomas Beth, Jörn Müller-Quade, and Rainer Steinwandt, Computing restrictions of ideals in finitely generated k-algebras by means of Buchberger's algorithm, J. Symbolic Comput. 41 (2006), no. 3-4, 372–380.[MR]
  10. Alin Bostan, Bruno Salvy, and Éric Schost, Fast algorithms for zero-dimensional polynomial systems using duality, Appl. Algebra Engrg. Comm. Comput. 14 (2003), no. 4, 239–272.[MR]
  11. Hoans-Christian Graf von Bothmer, Oliver Labs, Josef Schicho, and Christiaan van de Woestijne, The Casas-Alvero conjecture for infinitely many degrees, J. Algebra 316 (2007), no. 1, 224–230.[MR/link]
  12. Richard Brent and Paul Zimmermann, A multi-level blocking distinct degree factorization algorithm, Finite Fields and Applications, Contemporary Mathematics, vol. 461, 2008, 47–58 pages.
  13. Michael Brickenstein and Alexander Dreyer, PolyBoRi: a framework for Gröbner-basis computations with Boolean polynomials, J. Symbolic Comp. 44 (2009), no. 9, 1326–1345.
  14. Michael Brickenstein, Alexander Dreyer, Gert-Martin Greuel, Markus Wedler, and Oliver Wienand, New developments in the theory of Gröbner bases and applications to formal verification, J. Pure Appl. Algebra 213 (2009), no. 8, 1612–1635.[MR]
  15. Stanislav Bulygin and Ruud Pellikaan, Bounded distance decoding of linear error-correcting codes with Gröbner bases, J. Symb. Comput. 44 (2009), no. 12, 1626–1643.
  16. Daniel Cabarcas, An Implementation of Faugère's F4 Algorithm for Computing Gröbner Bases, Master's Thesis, University of Cincinnati, 2010.[link]
  17. G. Chèze and S. Najib, Indecomposability of polynomials via Jacobian matrix, J. Algebra 324 (2010), no. 1, 1–11.[MR/doi]
  18. Mihai Cipu, Gröbner bases and Diophantine analysis, J. Symbolic Comput. 43 (2008), no. 10, 681–687.[MR]
  19. Wolfram Decker and Theo de Jong, Gröbner bases and invariant theory, Gröbner bases and applications (Linz, 1998), London Math. Soc. Lecture Note Ser., vol. 251, Cambridge Univ. Press, Cambridge, 1998, pp. 61–89.[MR]
  20. Harm Derksen, Computation of invariants for reductive groups, Adv. Math. 141 (1999), no. 2, 366–384.[MR]
  21. Harm Derksen and Gregor Kemper, Computational Invariant Theory, Invariant Theory and Algebraic Transformation Groups, I, Springer-Verlag, Berlin, 2002, pp. x+268.[MR]
  22. Clémence Durvye and Grégoire Lecerf, A concise proof of the Kronecker polynomial system solver from scratch, Expo. Math. 26 (2008), no. 2, 101–139.[MR]
  23. Tobias Eibach, Enrico Pilz, and Gunnar Völkel, Attacking Bivium using SAT solvers, Theory and Applications of Satisfiability Testing, SAT 2008, Lecture Notes in Computer Science, vol. 4996, Springer, Berlin, 2008, pp. 63–76.
  24. Tobias Eibach, Gunnar Völkel, and Enrico Pilz, Optimising Gröbner bases on Bivium, Math. Comput. Sci. 3 (2010), no. 2, 159–172.[doi]
  25. Nicholas Eriksson, Toric ideals of homogeneous phylogenetic models, ISSAC 2004, ACM, New York, 2004, pp. 149–154.[MR/arXiv]
  26. Jeffrey B. Farr and Shuhong Gao, Computing Gröbner bases for vanishing ideals of finite sets of points, Applied Algebra, Algebraic Algorithms and Error-correcting Codes, Lecture Notes in Comput. Sci., vol. 3857, Springer, Berlin, 2006, pp. 118–127.[MR]
  27. Jeffrey B. Farr and Shuhong Gao, Gröbner bases and generalized Padé approximation, Math. Comp. 75 (2006), no. 253, 461–473 (electronic).[MR]
  28. Jean-Charles Faugère, Guillaume Moroz, Fabrice Rouillier, and Mohab Safey El Din, Classification of the perspective-three-point problem, discriminant variety and real solving polynomial systems of inequalities, in ISSAC '08: International Symposium on Symbolic and Algebraic Computation, ACM, New York, NY, USA, 2008, pp. 79–86.[doi]
  29. Akpodigha Filatei, Implementation of fast polynomial arithmentic in Aldor, Master's Thesis, University of Western Ontario, 2006.
  30. Shuhong Gao, Daqing Wan, and Mingsheng Wang, Primary decomposition of zero-dimensional ideals over finite fields, Math. Comp. 78 (2009), no. 265, 509–521.[MR/link]
  31. Karin Gatermann, Computer algebra methods for equivariant dynamical systems, Lecture Notes in Mathematics, vol. 1728, Springer-Verlag, Berlin, 2000, pp. xvi+153.[MR]
  32. Karin Gatermann and Frédéric Guyard, Gröbner bases, invariant theory and equivariant dynamics, J. Symbolic Comput. 28 (1999), no. 1-2, 275–302.[MR]
  33. V. P. Gerdt and Yu. A. Blinkov, On selection of nonmultiplicative prolongations in computation of Janet bases, Programming and Computer Software 33 (2007), no. 3, 147–153.
  34. V. P. Gerdt and Yu. A. Blinkov, Strategies for selecting non-multiplicative prolongations in computing Janet bases, Programmirovanie (2007), no. 3, 34–43.[MR]
  35. Vladimir P. Gerdt, Involutive algorithms for computing Gröbner bases, Computational Commutative and Non-commutative Algebraic Geometry, NATO Sci. Ser. III Comput. Syst. Sci., vol. 196, IOS, Amsterdam, 2005, pp. 199–225.[MR/link]
  36. Vladimir P. Gerdt and Yuri A. Blinkov, On computing Janet bases for degree compatible orderings, Proceedings of the 10th Rhine Workshop on Computer Algebra (Basel), 2006, University of Basel, Basel, 2006, pp. 107–117.
  37. Massimo Giulietti, Inviluppi di k-archi in piani proiettivi sopra campi finiti e basi di Gröbner, Rendiconti del Circolo Matematico di Palermo 48 (1999), no. 1, 191–200.
  38. Marc Giusti, Grégoire Lecerf, and Bruno Salvy, A Gröbner free alternative for polynomial system solving, J. Complexity 17 (2001), no. 1, 154–211.[MR]
  39. Marc Giusti and Éric Schost, Solving some overdetermined polynomial systems, in ISSAC '99: Proceedings of the 1999 International Symposium on Symbolic and Algebraic Computation (Vancouver, BC), ACM, New York, 1999, pp. 1–8 (electronic).[MR]
  40. Gert-Martin Greuel, Santiago Laplagne, and Frank Seelisch, Normalization of rings, J. Symbolic Comput. 45 (2010), no. 9, 887–901.[doi/arXiv]
  41. Renault Guénaél and Yokoyama Kazuhiro, Multi-modular algorithm for computing the splitting field of a polynomial, in ISSAC '08: International Symposium on Symbolic and Algebraic Computation, ACM, New York, NY, USA, 2008, pp. 247–254.[doi]
  42. David Harvey, A cache-friendly truncated FFT, Theor. Comput. Sci. 410 (2009), no. 27-29, 2649–2658.[arXiv]
  43. David Harvey, Faster polynomial multiplication via multipoint Kronecker substitution, J. Symbolic Comp. 44 (2009), no. 10, 1502–1510.
  44. Mark van Hoeij, Factoring polynomials and the knapsack problem, J. Number Theory 95 (2002), no. 2, 167–189.[MR]
  45. Mikael Johansson, Computation of Poincaré-Betti series for monomial rings, Rend. Istit. Mat. Univ. Trieste 37 (2005), no. 1-2, 85–94 (2006).[MR]
  46. Gregor Kemper, Computational invariant theory, The Curves Seminar at Queen's. Vol. XII (Kingston, ON, 1998), Queen's Papers in Pure and Appl. Math., vol. 114, Queen's Univ., Kingston, ON, 1998, pp. 5–26.[MR]
  47. Gregor Kemper, An algorithm to calculate optimal homogeneous systems of parameters, J. Symbolic Comput. 27 (1999), no. 2, 171–184.[MR]
  48. Gregor Kemper, The calculation of radical ideals in positive characteristic, J. Symbolic Comput. 34 (2002), no. 3, 229–238.[MR]
  49. Gregor Kemper, Computing invariants of reductive groups in positive characteristic, Transform. Groups 8 (2003), no. 2, 159–176.[MR]
  50. Simon King, Fast computation of secondary invariants, preprint (2007), 13 pages.[arXiv]
  51. Simon King, Minimal generating sets of non-modular invariant rings of finite groups, preprint (2007), 14 pages.[arXiv]
  52. Jennifer de Kleine, Michael Monagan, and Allan Wittkopf, Algorithms for the non-monic case of the sparse modular GCD algorithm, Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation: ISSAC'05, ACM, New York, 2005, pp. 124–131 (electronic).[MR]
  53. Alexey Koloydenko, Symmetric measures via moments, Bernoulli 14 (2008), no. 2, 362-390.[arXiv]
  54. Teresa Krick, Straight-line programs in polynomial equation solving, Foundations of Computational Mathematics: Minneapolis, 2002, London Math. Soc. Lecture Note Ser., vol. 312, Cambridge Univ. Press, Cambridge, 2004, pp. 96–136.[MR]
  55. G. Lecerf, Quadratic Newton iteration for systems with multiplicity, Found. Comput. Math. 2 (2002), no. 3, 247–293.[MR]
  56. Grégoire Lecerf, Computing the equidimensional decomposition of an algebraic closed set by means of lifting fibers, J. Complexity 19 (2003), no. 4, 564–596.[MR]
  57. Grégoire Lecerf, Fast separable factorization and applications, Appl. Algebra Engrg. Comm. Comput. 19 (2008), no. 2, 135–160.[MR]
  58. Grégoire Lecerf, New recombination algorithms for bivariate polynomial factorization based on Hensel lifting, Appl. Algebra Engrg. Comm. Comput. 21 (2010), no. 2, 151–176.[MR/doi]
  59. Xin Li, Marc Moreno Maza, Raqeeb Rasheed, and Eric Schost, High-performance symbolic computation in a hybrid compiled-interpreted programming environment, International Conference on Computational Sciences and Its Applications. ICCSA. June 30- July 3, 2008, 2008, pp. 331–341.
  60. Xin Li, Marc Moreno Maza, and Éric Schost, Fast arithmetic for triangular sets: from theory to practice, ISSAC 2007, ACM, New York, 2007, pp. 269–276.[MR]
  61. Xin Li, Marc Moreno Maza, and Éric Schost, Fast arithmetic for triangular sets: from theory to practice, J. Symbolic Comput. 44 (2009), no. 7, 891–907.[MR]
  62. A. Marschner and J. Müller, On a certain algebra of higher modular forms, Algebra Colloq. 16 (2009), 371–380.
  63. Mbakop Guy Merlin, Eziente losung reeller polynomialer gleichungssysteme, PhD Thesis, Humboldt-Universität, Berlin, 1999.[link]
  64. V. A. Mityunin and E. V. Pankratiev, Parallel algorithms for Gröbner-basis construction, J. Math. Sci. (N. Y.) 142 (2007), no. 4, 2248–2266.
  65. Michael Monagan and Mark van Hoeij, A modular algorithm for computing polynomial GCDs over number fields presented with multiple extensions, preprint http://www.cecm.sfu.ca/CAG/papers/HoeijMonGCD.pdf, 36 pages.
  66. Teo Mora, The FGLM problem and Möller's algorithm on zero-dimensional ideals, Sala, Massimiliano (ed.) and Mora, Teo (ed.) and Perret, Ludovic (ed.) and Sakata, Shojiro (ed.) and Traverso, Carlo (ed.), Gröbner Bases, Coding, and Cryptography, Springer, Berlin, 2009.
  67. Marc Moreno Maza, Greg Reid, Robin Scott, and Wenyuan Wu, On approximate triangular decompositions in dimension zero, J. Symbolic Comput. 42 (2007), no. 7, 693–716.[MR]
  68. Bernard Mourrain, Generalized normal forms and polynomial system solving, ISSAC'05: Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2005, pp. 253–260 (electronic).[MR]
  69. Bernard Mourrain and Philippe Trébuchet, Stable normal forms for polynomial system solving, Theoret. Comput. Sci. 409 (2008), no. 2, 229–240.[MR/doi]
  70. Jörn Müller-Quade and Rainer Steinwandt, Basic algorithms for rational function fields, J. Symbolic Comput. 27 (1999), no. 2, 143–170.[MR]
  71. Jörn Müller-Quade and Rainer Steinwandt, Gröbner bases applied to finitely generated field extensions, J. Symbolic Comput. 30 (2000), no. 4, 469–490.[MR]
  72. G. H. Norton and A. Salagean, Cyclic codes and minimal strong Gröbner bases over a principal ideal ring, Finite Fields Appl. 9 (2003), no. 2, 237–249.[MR/doi]
  73. Graham H. Norton and Ana Sălăgean, Strong Gröbner bases for polynomials over a principal ideal ring, Bull. Austral. Math. Soc. 64 (2001), no. 3, 505–528.[MR]
  74. Daniel Robertz, Noether normalization guided by monomial cone decompositions, J. Symbolic Comput. 44 (2009), no. 10, 1359–1373.[MR/doi]
  75. Fabrice Rouillier, Mohab Safey El Din, and Éric Schost, Solving the Birkhoff interpolation problem via the critical point method: an experimental study, ADG '00: Revised Papers from the Third International Workshop on Automated Deduction in Geometry (Zurich, 2000), Lecture Notes in Computer Science, vol. 2061, Springer-Verlag, Berlin, 2001, pp. viii+325.[MR]
  76. Luciano Sbaiz, Patrick Vandewalle, and Martin Vetterli, Groebner basis methods for multichannel sampling with unknown offsets, Appl. Comput. Harmon. Anal. 25 (2008), no. 3, 277–294.[doi]
  77. Roberto La Scala and Viktor Levandovskyy, Letterplace ideals and non-commutative Gröbner bases, J. Symbolic Comp. 44 (2009), no. 10, 1374-1393.
  78. Éric Schost, Degree bounds and lifting techniques for triangular sets, in Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2002, pp. 238–245 (electronic).[MR]
  79. Éric Schost, Complexity results for triangular sets, J. Symbolic Comput. 36 (2003), no. 3-4, 555–594.[MR]
  80. Éric Schost, Computing parametric geometric resolutions, Appl. Algebra Engrg. Comm. Comput. 13 (2003), no. 5, 349–393.[MR]
  81. R. James Shank and David L. Wehlau, Computing modular invariants of p-groups, J. Symbolic Comput. 34 (2002), no. 5, 307–327.[MR]
  82. Jessica Sidman and Seth Sullivant, Prolongations and computational algebra, Canad. J. Math. 61 (2009), no. 4, 930–949.[MR/link]
  83. Allan Steel, Conquering inseparability: Primary decomposition and multivariate factorization over algebraic function fields of positive characteristic, J. Symbolic Comput. 40 (2005), no. 3, 1053–1075.[MR]
  84. Till Stegers, Faugère's F5 algorithm revisited, Master's Thesis, Technische Universiteit Darmstadt, 2005.[link]
  85. Rainer Steinwandt, Decomposing systems of polynomial equations, Computer Algebra in Scientific Computing—CASC'99 (Munich), Springer, Berlin, 1999, pp. 387–407.[MR]
  86. Rainer Steinwandt, Implicitizing without tag variables, Proceedings of the 8th Rhine Workshop on Computer Algebra, 2002, pp. 217-224.
  87. Rainer Steinwandt and Jörn Müller-Quade, Freeness, linear disjointness, and implicitization—a classical approach, Beiträge Algebra Geom. 41 (2000), no. 1, 57–66.[MR]
  88. Pawel Wocjan, Brill-Noether algorithm construction of geometric Goppa codes and absolute factorization of polynomials, PhD Thesis, Institut für Algorithmen und Kognitive Systeme, Universität Karlsruhe, 1999.