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6 CONTENTS D Index of Notation 241
List of Figures 2.1 A polynomial SLP . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.2 Code fragment A — a graph . . . . . . . . . . . . . . . . . . . . 39 2.3 Code fragment B — a tree . . . . . . . . . . . . . . . . . . . . . . 39 2.4 DAG representation . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.5 Tree representation . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.1 Program for computing solutions to a cubic . . . . . . . . . . . . 57 3.2 Program for computing solutions to a quartic . . . . . . . . . . . 58 3.3 Cylindrical Deccomposition after Collins . . . . . . . . . . . . . . 109 4.1 Diagrammatic illustration of Modular Algorithms . . . . . . . . . 113 4.2 Diagrammatic illustration of Algorithm 13 . . . . . . . . . . . . . 118 4.3 Algorithm 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.4 Diagrammatic illustration of Algorithm 14 . . . . . . . . . . . . . 121 4.5 “Early termination” g.c.d. code . . . . . . . . . . . . . . . . . . . 122 4.6 Algorithm 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.7 Diagrammatic illustration of Algorithm 17 . . . . . . . . . . . . . 128 4.8 Algorithm 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.9 Diagrammatic illustration of g.c.d.s in Z[x, y] (1) . . . . . . . . . 130 4.10 Diagrammatic illustration of g.c.d.s in Z[x, y] (2) . . . . . . . . . 130 4.11 Diagrammatic illustration of sparse g.c.d. . . . . . . . . . . . . . 135 4.12 Algorithm 18: Sparse g.c.d. . . . . . . . . . . . . . . . . . . . . . 136 4.13 Algorithm 19: Inner sparse g.c.d. . . . . . . . . . . . . . . . . . . 136 4.14 Algorithm 20: Sparse g.c.d. from skeleton . . . . . . . . . . . . . 137 4.15 f from section 4.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.16 g from section 4.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 138 4.17 Algorithm 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 5.1 Diagrammatic illustration of Hensel Algorithms . . . . . . . . . . 149 5.2 Algorithm23: Distinct Degree Factorization . . . . . . . . . . . . 152 5.3 Algorithm24: Split a Distinct Degree Factorization . . . . . . . . 153 5.4 Algorithm 25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 5.5 Algorithm 26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 5.6 Algorithm 27 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 7
Page 1 and 2: Contents 1 Introduction 13 1.1 Hist
Page 3 and 4: CONTENTS 3 4 Modular Methods 113 4.
Page 5: CONTENTS 5 A Algebraic Background 2
Page 9 and 10: LIST OF FIGURES 9 List of Open Prob
Page 11 and 12: Preface This text is under active d
Page 13 and 14: Chapter 1 Introduction Computer alg
Page 15 and 16: 1.1. HISTORY AND SYSTEMS 15 1.1 His
Page 17 and 18: 1.2. EXPANSION AND SIMPLIFICATION 1
Page 23 and 24: 1.3. ALGEBRAIC DEFINITIONS 23 which
Page 25 and 26: 1.3. ALGEBRAIC DEFINITIONS 25 Propo
Page 27 and 28: 1.5. SOME MAPLE 27 Definition 18 (F
Page 29 and 30: Chapter 2 Polynomials Polynomials a
Page 31 and 32: 2.1. WHAT ARE POLYNOMIALS? 31 2.1.1
Page 33 and 34: 2.1. WHAT ARE POLYNOMIALS? 33 MULTI
Page 35 and 36: 2.1. WHAT ARE POLYNOMIALS? 35 not n
Page 37 and 38: 2.1. WHAT ARE POLYNOMIALS? 37 While
Page 39 and 40: 2.1. WHAT ARE POLYNOMIALS? 39 p:=x+
Page 41 and 42: 2.2. RATIONAL FUNCTIONS 41 Proposit
Page 43 and 44: 2.3. GREATEST COMMON DIVISORS 43 2.
Page 45 and 46: 2.3. GREATEST COMMON DIVISORS 45 Le
Page 47 and 48: 2.3. GREATEST COMMON DIVISORS 47 93
Page 49 and 50: 2.3. GREATEST COMMON DIVISORS 49 a
Page 51 and 52: 2.3. GREATEST COMMON DIVISORS 51 #
Page 53 and 54: 2.4. NON-COMMUTATIVE POLYNOMIALS 53
Page 55 and 56: 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
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8.2. BRANCH CUTS 199 8.2 Branch Cut
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8.2. BRANCH CUTS 201 Note that the
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8.5. INTEGRATING ‘REAL’ FUNCTIO
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Appendix A Algebraic Background A.1
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A.1. THE RESULTANT 207 Proposition
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A.2. USEFUL ESTIMATES 209 Corollary
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A.2. USEFUL ESTIMATES 211 Propositi
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A.2. USEFUL ESTIMATES 213 centring
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A.3. CHINESE REMAINDER THEOREM 215
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A.5. VANDERMONDE SYSTEMS 217 Clearl
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A.5. VANDERMONDE SYSTEMS 219 wherea
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Appendix B Excursus This appendix i
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B.2. EQUALITY OF FACTORED POLYNOMIA
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B.3. KARATSUBA’S METHOD 225 addit
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B.4. STRASSEN’S METHOD 227 answer
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B.4. STRASSEN’S METHOD 229 If the
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Appendix C Systems This appendix di
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C.2. MACSYMA 233 (1) -> a:=1 Figure
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C.3. MAPLE 235 > (x^2-1)^10/(x+1)^1
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C.3. MAPLE 237 Table C.2: Another s
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C.5. REDUCE 239 1: (x^2-1)^10/(x+1)
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Appendix D Index of Notation Notati
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Bibliography [Abb02] J.A. Abbott. S
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BIBLIOGRAPHY 245 [BCM94] [BCR98] [B
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BIBLIOGRAPHY 247 [Buc79] [Buc84] [B
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BIBLIOGRAPHY 249 [CW90] D. Coppersm
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BIBLIOGRAPHY 251 [FGT01] E. Fortuna
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BIBLIOGRAPHY 253 [Isa85] I.M. Isaac
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BIBLIOGRAPHY 255 [Loo82] R. Loos. G
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BIBLIOGRAPHY 257 [Per09] [PQR09] [P
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BIBLIOGRAPHY 259 [SS11] J. Schicho
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BIBLIOGRAPHY 261 [Zip79b] R.E. Zipp
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INDEX 263 Budan-Fourier theorem, 63
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INDEX 265 Polynomial, 186 Least com
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INDEX 267 Toeplitz Matrix, 65 Trage
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