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250 BIBLIOGRAPHY [DL08] J.H. Davenport and P. Libbrecht. The Freedom to Extend OpenMath and its Utility. Mathematics in Computer Science 2(2008/9), pages 379–398, 2008. [DM90] [DN07] [Dod66] [Doy99] [DS97] [DS00] [DT81] [DT90] [Duv87] [Fat03] [Fau02] [FGK + 94] [FGLM93] J.H. Davenport and M. Mignotte. On Finding the Largest Root of a Polynomial. Modélisation Mathématique et Analyse Numérique, 24:693–696, 1990. P. D’Alberto and A. Nicolau. Adaptive Strassen’s matrix multiplication. In Proceedings Supercomputing 2007, pages 284–292, 2007. C.L. Dodgson. Condensation of determinants, being a new and brief method for computing their algebraic value. Proc. Roy. Soc. Ser. A, 15:150–155, 1866. N.J. Doye. Automated Coercion for Axiom. In S. Dooley, editor, Proceedings ISSAC ’99, pages 229–235, 1999. A. Dolzmann and Th. Sturm. Redlog: Computer Algebra Meets Computer Logic. ACM SIGSAM Bull. 2, 31:2–9, 1997. J.H. Davenport and G.C. Smith. Fast recognition of alternating and symmetric groups. J. Pure Appl. Algebra, 153:17–25, 2000. J.H. Davenport and B.M. Trager. Factorization over finitely generated fields. In Proceedings SYMSAC 81, pages 200–205, 1981. J.H. Davenport and B.M. Trager. Scratchpad’s View of Algebra I: Basic Commutative Algebra. In Proceedings DISCO ’90, 1990. D. Duval. Diverses Questions relatives au Calcul Formel avec les Nombres Algébriques. Thèse d’Etat, 1987. R.J. Fateman. Comparing the speed of programs for sparse polynomial multiplication. SIGSAM Bulletin 1, 37:4–15, 2003. J.-C. Faugère. A New Efficient Algorithm for Computing Gröbner Bases Without Reduction to Zero (F 5 ). In T. Mora, editor, Proceedings ISSAC 2002, pages 75–83, 2002. B. Fuchssteiner, K. Gottheil, A. Kemper, O. Kluge, K. Morisse, H. Naundorf, G. Oevel, T. Schulze, and W. Wiwianka. Mu- PAD Multi Processing Algebra Data Tool Tutorial (version 1.2). Birkhäuser Verlag, 1994. J.C. Faugère, P. Gianni, D. Lazard, and T. Mora. Efficient Computation of Zero-Dimensional Gröbner Bases by Change of Ordering. J. Symbolic Comp., 16:329–344, 1993.
BIBLIOGRAPHY 251 [FGT01] E. Fortuna, P. Gianni, and B. Trager. Degree reduction under specialization. J. Pure Appl. Algebra, 164:153–163, 2001. [fHN76] J.P. ffitch, P. Herbert, and A.C. Norman. Design Features of COBALG. In R.D. Jenks, editor, Proceedings SYMSAC 76, pages 185–188, 1976. [FM89] D.J. Ford and J. McKay. Computation of Galois Groups from Polynomials over the Rationals. Computer Algebra (Lecture Notes in Pure and Applied Mathematics 113, 1989. [Fou31] J. Fourier. Analyse des équations déterminées. Didot, 1831. [FS56] [Ful69] A. Fröhlich and J.C. Shepherdson. Effective Procedures in Field Theory. Phil. Trans. Roy. Soc. Ser. A 248(1955-6), pages 407– 432, 1956. W. Fulton. Algebraic Curves, An Introduction to Algebraic Geometry. W.A. Benjamin Inc, 1969. [Gal79] E. Galois. Œuvres mathématiques. Gauthier-Villars (sous l’auspices de la SMF), 1879. [Gia89] P. Gianni. Properties of Gröbner bases under specializations. In Proceedings EUROCAL 87, pages 293–297, 1989. [GJY75] J.H. Griesmer, R.D. Jenks, and D.Y.Y. Yun. SCRATCHPAD User’s Manual. IBM Research Publication RA70, 1975. [GKP94] [GMN + 91] R.L. Graham, D.E. Knuth, and O. Patashnik. Concrete Mathematics (2nd edition). Addison-Wesley, 1994. A. Giovini, T. Mora, G. Niesi, L. Robbiano, and C. Traverso. One sugar cube, please, or selection strategies in the Buchberger algorithm. In S.M. Watt, editor, Proceedings ISSAC 1991, pages 49–54, 1991. [GN90] A. Giovini and G. Niesi. CoCoA: A User-Friendly System for Commutative Algebra. In Proceedings DISCO ’90, 1990. [Gos78] [Gra37] [HA10] [Ham07] Jr. Gosper, R.W. A Decision Procedure for Indefinite Hypergeometric Summation. Proc. Nat. Acad. Sci., 75:40–42, 1978. C.H. Graeffe. Die Auflösulg der höheren numerischen Gleichungen. F. Schulthess, 1837. A. Hashemi and G. Ars. Extended F5 criteria. J. Symbolic Comp., 45:1330–1340, 2010. S. Hammarling. Life as a developer of numerical software. Talk at NAG Ltd AGM, 2007.
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Contents 1 Introduction 13 1.1 Hist
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CONTENTS 3 4 Modular Methods 113 4.
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CONTENTS 5 A Algebraic Background 2
Page 7 and 8:
List of Figures 2.1 A polynomial SL
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LIST OF FIGURES 9 List of Open Prob
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Preface This text is under active d
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Chapter 1 Introduction Computer alg
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1.1. HISTORY AND SYSTEMS 15 1.1 His
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1.2. EXPANSION AND SIMPLIFICATION 1
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1.3. ALGEBRAIC DEFINITIONS 23 which
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1.3. ALGEBRAIC DEFINITIONS 25 Propo
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1.5. SOME MAPLE 27 Definition 18 (F
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Chapter 2 Polynomials Polynomials a
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2.1. WHAT ARE POLYNOMIALS? 31 2.1.1
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2.1. WHAT ARE POLYNOMIALS? 33 MULTI
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2.1. WHAT ARE POLYNOMIALS? 35 not n
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2.1. WHAT ARE POLYNOMIALS? 37 While
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2.1. WHAT ARE POLYNOMIALS? 39 p:=x+
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2.2. RATIONAL FUNCTIONS 41 Proposit
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2.3. GREATEST COMMON DIVISORS 43 2.
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2.3. GREATEST COMMON DIVISORS 45 Le
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2.3. GREATEST COMMON DIVISORS 47 93
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2.3. GREATEST COMMON DIVISORS 49 a
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2.3. GREATEST COMMON DIVISORS 51 #
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2.4. NON-COMMUTATIVE POLYNOMIALS 53
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Chapter 3 Polynomial Equations In t
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3.1. EQUATIONS IN ONE VARIABLE 57 S
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3.1. EQUATIONS IN ONE VARIABLE 59 I
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3.1. EQUATIONS IN ONE VARIABLE 61 T
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3.1. EQUATIONS IN ONE VARIABLE 63 S
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3.2. LINEAR EQUATIONS IN SEVERAL VA
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3.3. NONLINEAR MULTIVARIATE EQUATIO
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3.5. EQUATIONS AND INEQUALITIES 101
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3.6. CONCLUSIONS 111 Partial Proof.
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Chapter 4 Modular Methods In chapte
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4.1. GCD IN ONE VARIABLE 115 The co
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4.1. GCD IN ONE VARIABLE 117 This l
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4.1. GCD IN ONE VARIABLE 119 be mad
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4.1. GCD IN ONE VARIABLE 121 Figure
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4.1. GCD IN ONE VARIABLE 123 Figure
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4.2. POLYNOMIALS IN TWO VARIABLES 1
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4.3. POLYNOMIALS IN SEVERAL VARIABL
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4.4. FURTHER APPLICATIONS 139 2 7 3
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4.4. FURTHER APPLICATIONS 141 i.e.
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4.5. GRÖBNER BASES 143 Observation
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4.5. GRÖBNER BASES 145 Table 4.2:
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4.5. GRÖBNER BASES 147 Figure 4.17
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Chapter 5 p-adic Methods In this ch
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5.2. MODULAR METHODS 151 nomials, b
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5.4. FROM Z P TO Z? 153 Figure 5.3:
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5.5. HENSEL LIFTING 155 polynomials
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5.5. HENSEL LIFTING 157 Figure 5.4:
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5.5. HENSEL LIFTING 159 Figure 5.7:
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5.7. UNIVARIATE FACTORING SOLVED 16
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5.8. MULTIVARIATE FACTORING 163 Ope
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Chapter 6 Algebraic Numbers and fun
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6.1. THE D5 APPROACH TO ALGEBRAIC N
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Chapter 7 Calculus Throughout this
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7.2. INTEGRATION OF RATIONAL EXPRES
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7.3. THEORY: LIOUVILLE’S THEOREM
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7.4. INTEGRATION OF LOGARITHMIC EXP
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7.5. INTEGRATION OF EXPONENTIAL EXP
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7.7. THE RISCH DIFFERENTIAL EQUATIO
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7.8. THE PARALLEL APPROACH 193 i.e.
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7.10. OTHER CALCULUS PROBLEMS 195 7
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Chapter 8 Algebra versus Analysis W
Page 199 and 200: 8.2. BRANCH CUTS 199 8.2 Branch Cut
Page 201 and 202: 8.2. BRANCH CUTS 201 Note that the
Page 203 and 204: 8.5. INTEGRATING ‘REAL’ FUNCTIO
Page 205 and 206: Appendix A Algebraic Background A.1
Page 207 and 208: A.1. THE RESULTANT 207 Proposition
Page 209 and 210: A.2. USEFUL ESTIMATES 209 Corollary
Page 211 and 212: A.2. USEFUL ESTIMATES 211 Propositi
Page 213 and 214: A.2. USEFUL ESTIMATES 213 centring
Page 215 and 216: A.3. CHINESE REMAINDER THEOREM 215
Page 217 and 218: A.5. VANDERMONDE SYSTEMS 217 Clearl
Page 219 and 220: A.5. VANDERMONDE SYSTEMS 219 wherea
Page 221 and 222: Appendix B Excursus This appendix i
Page 223 and 224: B.2. EQUALITY OF FACTORED POLYNOMIA
Page 225 and 226: B.3. KARATSUBA’S METHOD 225 addit
Page 227 and 228: B.4. STRASSEN’S METHOD 227 answer
Page 229 and 230: B.4. STRASSEN’S METHOD 229 If the
Page 231 and 232: Appendix C Systems This appendix di
Page 233 and 234: C.2. MACSYMA 233 (1) -> a:=1 Figure
Page 235 and 236: C.3. MAPLE 235 > (x^2-1)^10/(x+1)^1
Page 237 and 238: C.3. MAPLE 237 Table C.2: Another s
Page 239 and 240: C.5. REDUCE 239 1: (x^2-1)^10/(x+1)
Page 241 and 242: Appendix D Index of Notation Notati
Page 243 and 244: Bibliography [Abb02] J.A. Abbott. S
Page 245 and 246: BIBLIOGRAPHY 245 [BCM94] [BCR98] [B
Page 247 and 248: BIBLIOGRAPHY 247 [Buc79] [Buc84] [B
Page 249: BIBLIOGRAPHY 249 [CW90] D. Coppersm
Page 253 and 254: BIBLIOGRAPHY 253 [Isa85] I.M. Isaac
Page 255 and 256: BIBLIOGRAPHY 255 [Loo82] R. Loos. G
Page 257 and 258: BIBLIOGRAPHY 257 [Per09] [PQR09] [P
Page 259 and 260: BIBLIOGRAPHY 259 [SS11] J. Schicho
Page 261 and 262: BIBLIOGRAPHY 261 [Zip79b] R.E. Zipp
Page 263 and 264: INDEX 263 Budan-Fourier theorem, 63
Page 265 and 266: INDEX 265 Polynomial, 186 Least com
Page 267 and 268: INDEX 267 Toeplitz Matrix, 65 Trage
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