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248 BIBLIOGRAPHY [CGG89] [CGGG83] B.W. Char, K.O. Geddes, and G.H. Gonnet. GCDHEU: Heuristic Polynomial GCD Algorithm Based on Integer GCD Computations. J. Symbolic Comp., 7:31–48, 1989. B.W. Char, K.O. Geddes, M.W. Gentleman, and G.H. Gonnet. The Design of MAPLE: A Compact, Portable and Powerful Computer Algebra System. In Proceedings EUROCAL 83 [Springer Lecture Notes in Computer Science 162, pages 101–115, 1983. [CGH88] L. Caniglia, A. Galligo, and J. Heintz. Some New Effectivity Bounds in Computational Geometry. In Proceedings AAECC-6, pages 131–152, 1988. [CGH + 03] [CH91] D. Castro, M. Giusti, J. Heintz, G. Matera, and L.M. Pardo. The Hardness of Polynomial Equation Solving. Foundations of Computational Mathematics, 3:347–420, 2003. G.E. Collins and H. Hong. Partial Cylindrical Algebraic Decomposition for Quantifier Elimination. J. Symbolic Comp., 12:299– 328, 1991. [CLO06] D.A. Cox, J.B. Little, and D.B. O’Shea. Ideals, Varieties and Algorithms. Springer–Verlag, 2006. [CMMXY09] C. Chen, M. Moreno Maza, B. Xia, and L. Yang. Computing Cylindrical Algebraic Decomposition via Triangular Decomposition. In J. May, editor, Proceedings ISSAC 2009, pages 95–102, 2009. [Col71] G.E. Collins. The SAC-1 System: An Introduction and Survey. In Proceedings SYMSAC 1971, 1971. [Col75] G.E. Collins. Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition. In Proceedings 2nd. GI Conference Automata Theory & Formal Languages, pages 134– 183, 1975. [Col79] [Col85] [Cor02] G.E. Collins. Factoring univariate integral polynomials in polynomial average time. In Proceedings EUROSAM 79, pages 317–329, 1979. G.E. Collins. The SAC-2 Computer Algebra System. In Proceedings EUROCAL 85, pages 34–35, 1985. R.M. Corless. Essential Maple 7 : an introduction for scientific programmers. Springer-Verlag, 2002. [CR88] M. Coste and M.F. Roy. Thom’s Lemma, the Coding of Real Algebraic Numbers and the Computation of the Topology of Semi- Algebraic Sets. J. Symbolic Comp., 5:121–129, 1988.
BIBLIOGRAPHY 249 [CW90] D. Coppersmith and S. Winograd. Matrix Multiplication via Arithmetic Progressions. J. Symbolic Comp., 9:251–280, 1990. [CZ81] [Dah09] D.G. Cantor and H. Zassenhaus. A New Algorithm for Factoring Polynomials over Finite Fields. Math. Comp., 36:587–592, 1981. X. Dahan. Size of coefficients of lexicographical Gröbner bases: the zero-dimensional, radical and bivariate case. In J. May, editor, Proceedings ISSAC 2009, pages 119–126, 2009. [Dav81] J.H. Davenport. On the Integration of Algebraic Functions. Springer Lecture Notes in Computer Science 102, 1981. [Dav84] J.H. Davenport. Intégration Algorithmique des fonctions élémentairement transcendantes sur une courbe algébrique. Annales de l’Institut Fourier, 34:271–276, 1984. [Dav85a] J.H. Davenport. Computer Algebra for Cylindrical Algebraic Decomposition. TRITA-NA-8511, 1985. [Dav85b] J.H. Davenport. The LISP/VM Foundations of Scratchpad II. Scratchpad II Newsletter 1, 1:4–5, 1985. [Dav97] [Dav10] J.H. Davenport. Galois Groups and the Simplification of Polynomials. Programming and Computer Software 1, 23:31–44, 1997. J.H. Davenport. The Challenges of Multivalued “Functions”. In S. Autexier et al., editor, Proceedings AISC/Calculemus/MKM 2010, pages 1–12, 2010. [DC10] J.H. Davenport and J. Carette. The Sparsity Challenges. In S. Watt et al., editor, Proceedings SYNASC 2009, pages 3–7, 2010. [DDDD85] J. Della Dora, C. DiCrescenzo, and D. Duval. About a new Method for Computing in Algebraic Number Fields. In Proceedings EUROCAL 85, pages 289–290, 1985. [DGT91] J.H. Davenport, P. Gianni, and B.M. Trager. Scratchpad’s View of Algebra II: A Categorical View of Factorization. In S.M. Watt, editor, Proceedings ISSAC 1991, pages 32–38, 1991. [DH88] J.H. Davenport and J. Heintz. Real Quantifier Elimination is Doubly Exponential. J. Symbolic Comp., 5:29–35, 1988. [Dic13] [DJ80] L.E. Dickson. Finiteness of the odd perfect and primitive abundant numbers with n prime factors. Amer. J. Math., 35:413–422, 1913. J.H. Davenport and R.D. Jenks. MODLISP — an Introduction. In Proceedings LISP80, 1980.
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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
Page 197 and 198: 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: BIBLIOGRAPHY 247 [Buc79] [Buc84] [B
Page 251 and 252: BIBLIOGRAPHY 251 [FGT01] E. Fortuna
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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