Contents - Student subdomain for University of Bath

More documents

Recommendations

Info

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 . . . . . . . . . . . . . . . . . . . . 82 3.3.6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.3.7 The Gianni–Kalkbrener Theorem . . . . . . . . . . . . . . 85 3.3.8 The Faugère–Gianni–Lazard–Mora Algorithm . . . . . . . 88 3.3.9 The Shape Lemma . . . . . . . . . . . . . . . . . . . . . . 90 3.3.10 The Hilbert function . . . . . . . . . . . . . . . . . . . . . 92 3.3.11 Coefficients other than fields . . . . . . . . . . . . . . . . 93 3.3.12 Non-commutative Ideals . . . . . . . . . . . . . . . . . . . 93 3.4 Nonlinear Multivariate Equations: Recursive . . . . . . . . . . . 94 3.4.1 Triangular Sets and Regular Chains . . . . . . . . . . . . 94 3.4.2 Positive Dimension . . . . . . . . . . . . . . . . . . . . . . 96 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 . . . . . . . . . . . . . . . . . . 103 3.5.4 Cylindrical Algebraic Decomposition . . . . . . . . . . . . 106 3.5.5 Computing Algebraic Decompositions . . . . . . . . . . . 108 3.5.6 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3.5.7 Further Observations . . . . . . . . . . . . . . . . . . . . . 110 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 #
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
Page 59 and 60:
3.1. EQUATIONS IN ONE VARIABLE 59 I
Page 61 and 62:
3.1. EQUATIONS IN ONE VARIABLE 61 T
Page 63 and 64:
3.1. EQUATIONS IN ONE VARIABLE 63 S
Page 65 and 66:
3.2. LINEAR EQUATIONS IN SEVERAL VA
Page 67 and 68:
Page 69 and 70:
Page 71 and 72:
3.3. NONLINEAR MULTIVARIATE EQUATIO
Page 73 and 74:
Page 75 and 76:
Page 77 and 78:
Page 79 and 80:
Page 81 and 82:
Page 83 and 84:
Page 85 and 86:
Page 87 and 88:
Page 89 and 90:
Page 91 and 92:
Page 93 and 94:
Page 95 and 96:
Page 97 and 98:
Page 99 and 100:
Page 101 and 102:
3.5. EQUATIONS AND INEQUALITIES 101
Page 103 and 104:
Page 105 and 106:
Page 107 and 108:
Page 109 and 110:
Page 111 and 112:
3.6. CONCLUSIONS 111 The process of
Page 113 and 114:
Chapter 4 Modular Methods In chapte
Page 115 and 116:
4.1. GCD IN ONE VARIABLE 115 The co
Page 117 and 118:
4.1. GCD IN ONE VARIABLE 117 This l
Page 119 and 120:
4.1. GCD IN ONE VARIABLE 119 be mad
Page 121 and 122:
4.1. GCD IN ONE VARIABLE 121 Figure
Page 123 and 124:
4.1. GCD IN ONE VARIABLE 123 Figure
Page 125 and 126:
4.2. POLYNOMIALS IN TWO VARIABLES 1
Page 127 and 128:
Page 129 and 130:
Page 131 and 132:
4.3. POLYNOMIALS IN SEVERAL VARIABL
Page 133 and 134:
Page 135 and 136:
Page 137 and 138:
Page 139 and 140:
4.4. FURTHER APPLICATIONS 139 2 7 3
Page 141 and 142:
4.4. FURTHER APPLICATIONS 141 i.e.
Page 143 and 144:
4.5. GRÖBNER BASES 143 Observation
Page 145 and 146:
4.5. GRÖBNER BASES 145 Table 4.2:
Page 147 and 148:
4.5. GRÖBNER BASES 147 Figure 4.17
Page 149 and 150:
Chapter 5 p-adic Methods In this ch
Page 151 and 152:
5.2. MODULAR METHODS 151 nomials, b
Page 153 and 154:
5.4. FROM Z P TO Z? 153 Figure 5.3:
Page 155 and 156:
5.5. HENSEL LIFTING 155 polynomials
Page 157 and 158:
5.5. HENSEL LIFTING 157 Figure 5.4:
Page 159 and 160:
5.5. HENSEL LIFTING 159 Figure 5.7:
Page 161 and 162:
5.7. UNIVARIATE FACTORING SOLVED 16
Page 163 and 164:
5.8. MULTIVARIATE FACTORING 163 Ope
Page 165 and 166:
Chapter 6 Algebraic Numbers and fun
Page 167 and 168:
6.1. THE D5 APPROACH TO ALGEBRAIC N
Page 169 and 170:
Chapter 7 Calculus Throughout this
Page 171 and 172:
7.2. INTEGRATION OF RATIONAL EXPRES
Page 173 and 174:
Page 175 and 176:
Page 177 and 178:
7.3. THEORY: LIOUVILLE’S THEOREM
Page 179 and 180:
Page 181 and 182:
Page 183 and 184:
7.4. INTEGRATION OF LOGARITHMIC EXP
Page 185 and 186:
7.5. INTEGRATION OF EXPONENTIAL EXP
Page 187 and 188:
Page 189 and 190:
Page 191 and 192:
7.7. THE RISCH DIFFERENTIAL EQUATIO
Page 193 and 194:
7.8. THE PARALLEL APPROACH 193 i.e.
Page 195 and 196:
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 and 248:
BIBLIOGRAPHY 247 [Buc79] [Buc84] [B
Page 249 and 250:
BIBLIOGRAPHY 249 [CW90] D. Coppersm
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 Calculus Fundamental Theo
Page 265 and 266:
INDEX 265 Lemma Frobenius, 229 Liou
Page 267 and 268:
INDEX 267 Triangular set, 95 Tschir
show all

Contents - Student subdomain for University of Bath

Create successful ePaper yourself

Delete template?

Save as template?