Fast computation of special resultants

Journal of Symbolic Computation

Volumn 41, Issue 1, 2006, Pages 1-29

  Bostan, Alin ^a   Flajolet, Philippe ^a   Salvy, Bruno ^a   Schost, Éric ^b  

^a INRIA ROCQUENCOURT   (France)

^b ÉCOLE POLYTECHNIQUE   (France)

Author keywords

Complexity; Composed product; Composed sum; Diamond product; Tellegen's principle

Indexed keywords

EID: 29644432637     PISSN: 07477171     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jsc.2005.07.001     Document Type: Article

References (81)

1. S.A. Abramov On the summation of rational functions Akademiya Nauk SSSR. Zhurnal Vychislitel′noǐMatematiki i MatematicheskoǐFiziki 11 4 1971 1071-1075
2. S.A. Abramov Rational solutions of linear differential and difference equations with polynomial coefficients Akademiya Nauk SSSR. Zhurnal Vychislitel′noǐMatematiki i MatematicheskoǐFiziki 29 11 1989 1611-1620,1757
3. A.V. Aho K. Steiglitz J.D. Ullman Evaluating polynomials at fixed sets of points SIAM Journal of Computing 4 4 1975 533-539
4. C. Banderier P. Flajolet Basic analytic combinatorics of directed lattice paths Theoretical Computer Science 281 1-2 2002 37-80
5. J. Berstel M. Mignotte Deux propriétés décidables des suites récurrentes linéaires Bulletin de la Société Mathématique de France 104 2 1976 175-184
6. D. Bini V.Y. Pan Polynomial and Matrix Computations vol. 1 1994 Birkhäuser Boston Inc. Boston, MA
7. Bostan, A., Flajolet, P., Salvy, B., Schost, É, 2002. Fast computation with two algebraic numbers. Research Report 4579, Institut National de Recherche en Informatique et en Automatique, 20 pages
8. Bostan, A., González-Vega, L., Perdry, H., Schost, É., 2005. From Newton sums to coefficients: Complexity issues in characteristic p. In: MEGA'05
9. A. Bostan G. Lecerf É. Schost Tellegen's principle into practice ISSAC'03 2003 ACM Press 37-44
10. A. Bostan B. Salvy É. Schost Fast algorithms for zero-dimensional polynomial systems using duality Applicable Algebra in Engineering, Communication and Computing 14 4 2003 239-272
11. R. Brandl Integer polynomials that are reducible modulo all primes The American Mathematical Monthly 93 4 1986 286-288
12. J.V. Brawley L. Carlitz Irreducibles and the composed product for polynomials over a finite field Discrete Mathematics 65 2 1987 115-139
13. J.V. Brawley S. Gao D. Mills Computing composed products of polynomials Finite Fields: Theory, Applications, and Algorithms (Waterloo, ON, 1997) Contemp. Math. vol. 225 1999 AMS 1-15
14. R.P. Brent Multiple-precision zero-finding methods and the complexity of elementary function evaluation Analytic Computational Complexity (Pittsburgh, 1975) 1976 Academic Press 151-176
15. R.P. Brent H.T. Kung Fast algorithms for manipulating formal power series Journal of the Association for Computing Machinery 25 4 1978 581-595
16. E. Briand L. González-Vega Multivariate Newton sums: Identities and generating functions Communications in Algebra 30 9 2002 4527-4547
17. P. Bürgisser M. Clausen M.A. Shokrollahi Algebraic Complexity Theory Grundlehren der Mathematischen Wissenschaften Fundamental Principles of Mathematical Sciences vol. 315 1997 Springer-Verlag Berlin
18. D.G. Cantor E. Kaltofen On fast multiplication of polynomials over arbitrary algebras Acta Informatica 28 7 1991 693-701
19. D. Casperson J. McKay Symmetric functions, m -sets, and Galois groups Mathematics of Computation 63 208 1994 749-757
20. H. Cohen A Course in Computational Algebraic Number Theory Graduate Texts in Mathematics vol. 138 1993 Springer-Verlag Berlin
21. D. Coppersmith S. Winograd Matrix multiplication via arithmetic progressions Journal of Symbolic Computation 9 3 1990 251-280
22. D. Cox J. Little D. O'Shea Using Algebraic Geometry Graduate Texts in Mathematics vol. 185 1998 Springer-Verlag New York
23. R. Dvornicich C. Traverso Newton symmetric functions and the arithmetic of algebraically closed fields AAECC-5 (Menorca, 1987) LNCS vol. 356 1989 Springer 216-224
24. P. Gaudry R. Harley Counting points on hyperelliptic curves over finite fields Proceedings of ANTS IV LNCS vol. 1838 2000 Springer-Verlag 313-332
25. P. Gaudry É. Schost Construction of secure random curves of genus 2 over prime fields Advances in Cryptology-EUROCRYPT 2004 LNCS vol. 3027 2004 Springer-Verlag 239-256
26. J. Gerhard M. Giesbrecht A. Storjohann E.V. Zima Shiftless decomposition and polynomial-time rational summation ISSAC'03 2003 ACM Press 119-126
27. M. Giusti D. Lazard A. Valibouze Algebraic transformations of polynomial equations, symmetric polynomials and elimination ISSAC'88 LNCS vol. 358 1989 Springer-Verlag 309-314
28. M.-J. Gonźlez-López L. González-Vega Newton identities in the multivariate case: Pham systems Gröbner Bases and Applications (Linz, 1998) London Mathematical Society Lecture Note Series vol. 251 1998 Cambridge University Press 351-366
29. L. González-Vega G. Trujillo Implicitization of parametric curves and surfaces by using symmetric functions ISSAC'95 1995 ACM Press 180-186
30. L. González-Vega G. Trujillo Using symmetric functions to describe the solution set of a zero-dimensional ideal AAECC-11 (Paris, 1995) LNCS vol. 948 1995 Springer 232-247
31. R.W. Gosper Jr. Decision procedure for indefinite hypergeometric summation Proceedings of the National Academy of Sciences of the United States of America 75 1 1978 40-42
32. G. Hanrot M. Quercia P. Zimmermann The middle product algorithm, I. Speeding up the division and square root of power series Applicable Algebra in Engineering, Communication and Computing 14 6 2004 415-438
33. P. Henrici Applied and Computational Complex Analysis vol. 3 1986 John Wiley & Sons Inc New York
34. X. Huang V.Y. Pan Fast rectangular matrix multiplication and applications Journal of Complexity 14 2 1998 257-299
35. E. Kaltofen Challenges of symbolic computation: My favorite open problems Journal of Symbolic Computation 29 6 2000 891-919
36. E. Kaltofen D.R. Musser B.D. Saunders A generalized class of polynomials that are hard to factor SIAM Journal on Computing 12 3 1983 473-483
37. E. Kaltofen V. Pan Processor-efficient parallel solution of linear systems II: The positive characteristic and singular cases FOCS'92 1992 IEEE Computer Society Press 714-723
38. E. Kaltofen V.Y. Pan Parallel solution of Toeplitz and Toeplitz-like linear systems over fields of small positive characteristic PASCO '94 Lecture Notes Ser. Comput vol. 5 1994 World Sci. Publishing 225-233
39. E. Kaltofen V. Shoup Subquadratic-time factoring of polynomials over finite fields Mathematics of Computation 67 223 1998 1179-1197
40. P. Kravanja M. Van Barel Coupled Vandermonde matrices and the superfast computation of Toeplitz determinants Numerical Algorithms 24 1-2 2000 99-116
41. L. Kronecker Grundzüge einer arithmetischen Theorie der algebraischen Grössen Journal für die reine und angewandte Mathematik 92 1882 1-122
42. H.T. Kung On computing reciprocals of power series Numerische Mathematik 22 1974 341-348
43. A. Lascoux La résultante de deux polynô In: P. Dubreil M.-P. Malliavin eds. Séminaire d'algèbre LNM vol. 1220 1986 Springer 56-72
44. U. Le Verrier Sur les variations séculaires des éléments elliptiques des sept planètes principales: Mercure, Vénus, La Terre, Mars, Jupiter, Saturne et Uranus Journal de Mathématiques Pures et Appliquées 4 1840 220-254
45. T. Lickteig M.-F. Roy Cauchy index computation Calcolo 33 3-4 1996 337-351
46. T. Lickteig M.-F. Roy Sylvester-Habicht sequences and fast Cauchy index computation Journal of Symbolic Computation 31 3 2001 315-341
47. R. Loos Computing in algebraic extensions Computer Algebra 1983 Springer 173-187
48. Y.-K. Man F. Wright Fast polynomial dispersion computation and its application to indefinite summation ISSAC'94 1994 ACM Press 175-180
49. V.Y. Pan Parallel computation of polynomial GCD and some related parallel computations over abstract fields Theoretical Computer Science 162 2 1996 173-223
50. V.Y. Pan Faster solution of the key equation for decoding BCH error-correcting codes Proceedings STOC'97 1997 ACM Press 168-175
51. V.Y. Pan New techniques for the computation of linear recurrence coefficients Finite Fields and their Applications 6 1 2000 93-118
52. M.S. Paterson L.J. Stockmeyer On the number of nonscalar multiplications necessary to evaluate polynomials SIAM Journal on Computing 2 1 1973 60-66
53. P. Paule Greatest factorial factorization and symbolic summation Journal of Symbolic Computation 20 3 1995 235-268
54. M. Petkovšek Hypergeometric solutions of linear recurrences with polynomial coefficients Journal of Symbolic Computation 14 2-3 1992 243-264
55. D. Reischert Asymptotically fast computation of subresultants ISSAC'97 1997 ACM Press 233-240
56. Rif̀, J., Borrell, J., 1991. Improving the time complexity of the computation of irreducible and primitive polynomials in finite fields, In: AAECC-9 (New Orleans, 1991), vol. 539, pp. 352-359
57. Rothstein, M., 1977. A new algorithm for the integration of exponential and logarithmic functions, In: Proceedings of the 1977 MACSYMA Users Conference, NASA Pub., CP-2012, pp. 263-274
58. F. Rouillier Solving zero-dimensional systems through the Rational Univariate Representation Applicable Algebra in Engineering, Communication and Computing 9 5 1999 433-461
59. A. Schönhage Schnelle Multiplikation von Polynomen über Körpern der Charakteristik 2 Acta Informatica 7 1977 395-398
60. Schönhage, A., 1982. The fundamental theorem of algebra in terms of computational complexity. Tech. Rep., Univ. Tübingen, 73 pages
61. A Schönhage Fast parallel computation of characteristic polynomials by Leverrier's power sum method adapted to fields of finite characteristic Automata, Languages and Programming (Lund, 1993) LNCS vol. 700 1993 Springer 410-417
62. A. Schönhage V. Strassen Schnelle Multiplikation großer Zahlen Computing 7 1971 281-292
63. R. Schoof Elliptic curves over finite fields and the computation of square roots mod p Mathematics of Computation 44 1985 483-494
64. Schost É. Multivariate power series multiplication 2005 293-300
65. J.T. Schwartz Fast probabilistic algorithms for verification of polynomial identities Journal of the Association for Computing Machinery 27 4 1980 701-717
66. V. Shoup New algorithms for finding irreducible polynomials over finite fields Mathematics of Computation 54 189 1990 435-447
67. V. Shoup Fast construction of irreducible polynomials over finite fields Journal of Symbolic Computation 17 5 1994 371-391
68. V. Shoup A new polynomial factorization algorithm and its implementation Journal of Symbolic Computation 20 4 1995 363-397
69. Shoup, V., 1996-2005. NTL: A library for doing number theory. http://www.shoup.net interref1
70. V. Shoup Efficient computation of minimal polynomials in algebraic extensions of finite fields ISSAC'99 1999 ACM Press 53-58
71. M. Sieveking An algorithm for division of powerseries Computing 10 1972 153-156
72. V. Strassen Gaussian elimination is not optimal Numerische Mathematik 13 1969 354-356
73. J.-A. Thiong Ly Note for computing the minimum polynomial of elements in large finite fields Coding Theory and Applications (Toulon, 1988) LNCS vol. 388 1989 Springer 185-192
74. B.M. Trager Algebraic factoring and rational function integration SYMSAC'76 1976 ACM Press 219-226
75. A. Valibouze Fonctions symétriques et changements de bases EUROCAL '87 (Leipzig, 1987) LNCS vol. 378 1989 Springer 323-332
76. J. von zur Gathen J. Gerhard Fast algorithms for Taylor shifts and certain difference equations ISSAC'97 1997 ACM Press 40-47
77. J. von zur Gathen J. Gerhard Modern Computer Algebra 1999 Cambridge University Press New York
78. F. von Haeseler W. Jürgensen Irreducible polynomials generated by decimations Finite Fields and Their Applications (Augsburg, 1999) 2001 Springer 224-231
79. M. van Hoeij Factoring polynomials and the knapsack problem Journal of Number Theory 95 2 2002 167-189
80. K. Yokoyama Z. Li I. Nemes Finding roots of unity among quotients of the roots of an integral polynomial ISSAC'95 1995 ACM Press 85-89
81. R. Zippel Effective Polynomial Computation 1993 Kluwer Academic Publishers Boston