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2 CONTENTS 3 Polynomial Equations 55 3.1 Equations in One Variable . . . . . . . . . . . . . . . . . . . . . . 55 3.1.1 Quadratic Equations . . . . . . . . . . . . . . . . . . . . . 55 3.1.2 Cubic Equations . . . . . . . . . . . . . . . . . . . . . . . 56 3.1.3 Quartic Equations . . . . . . . . . . . . . . . . . . . . . . 58 3.1.4 Higher Degree Equations . . . . . . . . . . . . . . . . . . 58 3.1.5 Reducible defining polynomials . . . . . . . . . . . . . . . 59 3.1.6 Multiple Algebraic Numbers . . . . . . . . . . . . . . . . . 60 3.1.7 Solutions in Real Radicals . . . . . . . . . . . . . . . . . . 60 3.1.8 Equations of curves . . . . . . . . . . . . . . . . . . . . . 61 3.1.9 How many real roots? . . . . . . . . . . . . . . . . . . . . 62 3.2 Linear Equations in Several Variables . . . . . . . . . . . . . . . 64 3.2.1 Linear Equations and Matrices . . . . . . . . . . . . . . . 64 3.2.2 Representations of Matrices . . . . . . . . . . . . . . . . . 65 3.2.3 Matrix Inverses: not a good idea! . . . . . . . . . . . . . . 66 3.2.4 Over/under-determined Systems . . . . . . . . . . . . . . 70 3.3 Nonlinear Multivariate Equations: Distributed . . . . . . . . . . 71 3.3.1 Gröbner Bases . . . . . . . . . . . . . . . . . . . . . . . . 74 3.3.2 How many Solutions? . . . . . . . . . . . . . . . . . . . . 76 3.3.3 Orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.3.4 Complexity of Gröbner Bases . . . . . . . . . . . . . . . . 80 3.3.5 A Matrix Formulation . . . . . . . . . . . . . . . . . . . . 83 3.3.6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.3.7 The Gianni–Kalkbrener Theorem . . . . . . . . . . . . . . 86 3.3.8 The Faugère–Gianni–Lazard–Mora Algorithm . . . . . . . 88 3.3.9 The Shape Lemma . . . . . . . . . . . . . . . . . . . . . . 91 3.3.10 The Hilbert function . . . . . . . . . . . . . . . . . . . . . 92 3.3.11 Coefficients other than fields . . . . . . . . . . . . . . . . 93 3.3.12 Non-commutative Ideals . . . . . . . . . . . . . . . . . . . 94 3.4 Nonlinear Multivariate Equations: Recursive . . . . . . . . . . . 95 3.4.1 Triangular Sets and Regular Chains . . . . . . . . . . . . 95 3.4.2 Positive Dimension . . . . . . . . . . . . . . . . . . . . . . 97 3.4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.5 Equations and Inequalities . . . . . . . . . . . . . . . . . . . . . . 100 3.5.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.5.2 Quantifier Elimination . . . . . . . . . . . . . . . . . . . . 102 3.5.3 Algebraic Decomposition . . . . . . . . . . . . . . . . . . 104 3.5.4 Cylindrical Algebraic Decomposition . . . . . . . . . . . . 106 3.5.5 Computing Algebraic Decompositions . . . . . . . . . . . 109 3.5.6 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . 110 3.5.7 Further Observations . . . . . . . . . . . . . . . . . . . . . 111 3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
CONTENTS 3 4 Modular Methods 113 4.1 Gcd in one variable . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.1.1 Bounds on divisors . . . . . . . . . . . . . . . . . . . . . . 115 4.1.2 The modular – integer relationship . . . . . . . . . . . . . 116 4.1.3 Computing the g.c.d.: one large prime . . . . . . . . . . . 118 4.1.4 Computing the g.c.d.: several small primes . . . . . . . . 120 4.1.5 Computing the g.c.d.: early success . . . . . . . . . . . . . 122 4.1.6 An alternative correctness check . . . . . . . . . . . . . . 122 4.1.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.2 Polynomials in two variables . . . . . . . . . . . . . . . . . . . . . 124 4.2.1 Degree Growth in Coefficients . . . . . . . . . . . . . . . . 124 4.2.2 The evaluation–interpolation relationship . . . . . . . . . 126 4.2.3 G.c.d. in Z p [x, y] . . . . . . . . . . . . . . . . . . . . . . . 127 4.2.4 G.c.d. in Z[x, y] . . . . . . . . . . . . . . . . . . . . . . . 128 4.3 Polynomials in several variables . . . . . . . . . . . . . . . . . . . 131 4.3.1 A worked example . . . . . . . . . . . . . . . . . . . . . . 132 4.3.2 Converting this to an algorithm . . . . . . . . . . . . . . . 134 4.3.3 Worked example continued . . . . . . . . . . . . . . . . . 135 4.3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 139 4.4 Further Applications . . . . . . . . . . . . . . . . . . . . . . . . . 139 4.4.1 Matrix Determinants . . . . . . . . . . . . . . . . . . . . . 139 4.4.2 Resultants and Discriminants . . . . . . . . . . . . . . . . 140 4.4.3 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . 140 4.5 Gröbner Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 4.5.1 General Considerations . . . . . . . . . . . . . . . . . . . 143 4.5.2 The Hilbert Function and reduction . . . . . . . . . . . . 144 4.5.3 The Modular Algorithm . . . . . . . . . . . . . . . . . . . 145 4.5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 146 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 5 p-adic Methods 149 5.1 Introduction to the factorization problem . . . . . . . . . . . . . 149 5.2 Modular methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 5.3 Factoring modulo a prime . . . . . . . . . . . . . . . . . . . . . . 151 5.3.1 Berlekamp’s small p method . . . . . . . . . . . . . . . . . 151 5.3.2 Berlekamp’s large p method . . . . . . . . . . . . . . . . . 151 5.3.3 The Cantor–Zassenhaus method . . . . . . . . . . . . . . 151 5.4 From Z p to Z? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 5.5 Hensel Lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 5.5.1 Linear Hensel Lifting . . . . . . . . . . . . . . . . . . . . . 155 5.5.2 Quadratic Hensel Lifting . . . . . . . . . . . . . . . . . . . 158 5.5.3 Hybrid Hensel Lifting . . . . . . . . . . . . . . . . . . . . 160 5.6 The recombination problem . . . . . . . . . . . . . . . . . . . . . 160 5.7 Univariate Factoring Solved . . . . . . . . . . . . . . . . . . . . . 161 5.8 Multivariate Factoring . . . . . . . . . . . . . . . . . . . . . . . . 163 5.9 Other Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Page 1: Contents 1 Introduction 13 1.1 Hist
Page 5 and 6: CONTENTS 5 A Algebraic Background 2
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 #
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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
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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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