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8 LIST OF FIGURES 5.7 Algorithm 28 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 5.8 Overview of Factoring Algorithm . . . . . . . . . . . . . . . . . . 162 6.1 Algebraic numbers in the denominator . . . . . . . . . . . . . . . 166 7.1 Algorithm 33: IntLog–Polynomial . . . . . . . . . . . . . . . . . . 184 7.2 Algorithm 34: IntLog–Rational Expression . . . . . . . . . . . . . 185 7.3 Algorithm 35: IntExp–Polynomial . . . . . . . . . . . . . . . . . 187 7.4 Algorithm 36: IntExp–Rational Expression . . . . . . . . . . . . 188 8.1 A Riemann surface example: log . . . . . . . . . . . . . . . . . . 200 C.1 Axiom output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 C.2 Axiom type system . . . . . . . . . . . . . . . . . . . . . . . . . . 233 C.3 Macsyma output . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 C.4 Maple output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 C.5 Tree for A, B corresponding to table C.1 . . . . . . . . . . . . . . 236 C.6 Tree for A, B corresponding to table C.2 . . . . . . . . . . . . . . 237 C.7 MuPAD output . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 C.8 Reduce output . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
LIST OF FIGURES 9 List of Open Problems 1 Roots of Sparse Polynomials . . . . . . . . . . . . . . . . . . . . . . 64 2 Sparse Gröbner Bases . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3 Complexity of the FGLM Algorithm . . . . . . . . . . . . . . . . . . 89 4 Improving Landau–Mignotte for g.c.d. . . . . . . . . . . . . . . . . . 119 5 Alternative Route for Bivariate Polynomial g.c.d. . . . . . . . . . . . 125 6 Which is the Better Route for Bivariate g.c.d.? . . . . . . . . . . . . 130 7 Contradictory Hilbert Functions . . . . . . . . . . . . . . . . . . . . 144 8 Bad Reduction for Gröbner Bases . . . . . . . . . . . . . . . . . . . 145 9 Modular Gröbner Bases for Inhomogeneous Ideals . . . . . . . . . . 146 10 Reconstructed Bases might not be Gröbner . . . . . . . . . . . . . . 146 11 Evaluate [vH02] against [ASZ00] . . . . . . . . . . . . . . . . . . . . 161 12 Better Choice of ‘Best’ Prime . . . . . . . . . . . . . . . . . . . . . . 163 13 Low-degree Factorization . . . . . . . . . . . . . . . . . . . . . . . . 163 14 Algebraic Numbers Reviewed . . . . . . . . . . . . . . . . . . . . . . 167 15 Real Roots of Sparse Polynomials . . . . . . . . . . . . . . . . . . . . 222 16 Crossings of factored polynomials . . . . . . . . . . . . . . . . . . . . 223
Page 1 and 2: Contents 1 Introduction 13 1.1 Hist
Page 3 and 4: CONTENTS 3 4 Modular Methods 113 4.
Page 5 and 6: CONTENTS 5 A Algebraic Background 2
Page 7: List of Figures 2.1 A polynomial SL
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
Page 57 and 58: 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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