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4 CONTENTS 5.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 6 Algebraic Numbers and functions 165 6.1 The D5 approach to algebraic numbers . . . . . . . . . . . . . . . 167 7 Calculus 169 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 7.2 Integration of Rational Expressions . . . . . . . . . . . . . . . . . 171 7.2.1 Integration of Proper Rational Expressions . . . . . . . . 171 7.2.2 Hermite’s Algorithm . . . . . . . . . . . . . . . . . . . . . 172 7.2.3 The Ostrogradski–Horowitz Algorithm . . . . . . . . . . . 173 7.2.4 The Trager–Rothstein Algorithm . . . . . . . . . . . . . . 174 7.3 Theory: Liouville’s Theorem . . . . . . . . . . . . . . . . . . . . . 176 7.3.1 Liouville’s Principle . . . . . . . . . . . . . . . . . . . . . 179 7.3.2 Finding L . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 7.3.3 Risch Structure Theorem . . . . . . . . . . . . . . . . . . 181 7.3.4 Overview of Integration . . . . . . . . . . . . . . . . . . . 181 7.4 Integration of Logarithmic Expressions . . . . . . . . . . . . . . . 182 7.4.1 The Polynomial Part . . . . . . . . . . . . . . . . . . . . . 183 7.4.2 The Rational Expression Part . . . . . . . . . . . . . . . . 184 7.5 Integration of Exponential Expressions . . . . . . . . . . . . . . . 184 7.5.1 The Polynomial Part . . . . . . . . . . . . . . . . . . . . . 186 7.5.2 The Rational Expression Part . . . . . . . . . . . . . . . . 187 7.6 Integration of Algebraic Expressions . . . . . . . . . . . . . . . . 190 7.7 The Risch Differential Equation Problem . . . . . . . . . . . . . 191 7.8 The Parallel Approach . . . . . . . . . . . . . . . . . . . . . . . . 193 7.8.1 The Parallel Approach: Algebraic Expressions . . . . . . 194 7.9 Definite Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 194 7.10 Other Calculus Problems . . . . . . . . . . . . . . . . . . . . . . 195 7.10.1 Indefinite summation . . . . . . . . . . . . . . . . . . . . . 195 7.10.2 Definite Symbolic Summation . . . . . . . . . . . . . . . . 195 7.10.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 7.10.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 8 Algebra versus Analysis 197 8.1 Functions and Formulae . . . . . . . . . . . . . . . . . . . . . . . 197 8.2 Branch Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 8.2.1 Removable Branch Cuts . . . . . . . . . . . . . . . . . . . 202 8.3 Fundamental Theorem of Calculus Revisited . . . . . . . . . . . . 202 8.4 Constants Revisited . . . . . . . . . . . . . . . . . . . . . . . . . 202 8.4.1 Constants can be useful . . . . . . . . . . . . . . . . . . . 202 8.5 Integrating ‘real’ Functions . . . . . . . . . . . . . . . . . . . . . 203 8.6 Logarithms revisited . . . . . . . . . . . . . . . . . . . . . . . . . 203
CONTENTS 5 A Algebraic Background 205 A.1 The resultant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 A.2 Useful Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 A.2.1 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 A.2.2 Coefficients of a polynomial . . . . . . . . . . . . . . . . . 209 A.2.3 Roots of a polynomial . . . . . . . . . . . . . . . . . . . . 211 A.2.4 Root separation . . . . . . . . . . . . . . . . . . . . . . . . 212 A.3 Chinese Remainder Theorem . . . . . . . . . . . . . . . . . . . . 215 A.4 Chinese Remainder Theorem for Polynomials . . . . . . . . . . . 216 A.5 Vandermonde Systems . . . . . . . . . . . . . . . . . . . . . . . . 217 B Excursus 221 B.1 The Budan–Fourier Theorem . . . . . . . . . . . . . . . . . . . . 221 B.2 Equality of factored polynomials . . . . . . . . . . . . . . . . . . 222 B.3 Karatsuba’s method . . . . . . . . . . . . . . . . . . . . . . . . . 224 B.3.1 Karatsuba’s method in practice . . . . . . . . . . . . . . . 225 B.3.2 Karatsuba’s method and sparse polynomials . . . . . . . . 226 B.3.3 Karatsuba’s method and multivariate polynomials . . . . 226 B.3.4 Faster still . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 B.3.5 Faster division . . . . . . . . . . . . . . . . . . . . . . . . 227 B.3.6 Faster g.c.d. computation . . . . . . . . . . . . . . . . . . 227 B.4 Strassen’s method . . . . . . . . . . . . . . . . . . . . . . . . . . 227 B.4.1 Strassen’s method in practice . . . . . . . . . . . . . . . . 229 B.4.2 Further developments . . . . . . . . . . . . . . . . . . . . 229 B.4.3 Matrix Inversion . . . . . . . . . . . . . . . . . . . . . . . 229 C Systems 231 C.1 Axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 C.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 C.1.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 C.1.3 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 C.2 Macsyma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 C.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 C.2.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 C.3 Maple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 C.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 C.3.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 C.3.3 Data structures . . . . . . . . . . . . . . . . . . . . . . . . 234 C.3.4 Heuristic GCD . . . . . . . . . . . . . . . . . . . . . . . . 237 C.3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 237 C.4 MuPAD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 C.4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 C.4.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 C.5 Reduce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 C.5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 C.5.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
Page 1 and 2: Contents 1 Introduction 13 1.1 Hist
Page 3: CONTENTS 3 4 Modular Methods 113 4.
Page 7 and 8: List of Figures 2.1 A polynomial SL
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
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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
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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
Page 267 and 268:
INDEX 267 Toeplitz Matrix, 65 Trage
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