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264 INDEX Excel, 13 Existential theory of the reals, 110 Exponentiation polynomial, 30 Expression DAG representation, 38 tree representation, 37 Farey fractions, 141 reconstruction, 141 Faugère–Gianni–Lazard–Mora algorithm, 88 Field, 25 difference, 195 differential, 169 of fractions, 25 real closed, 101 Fill-in (loss of sparsity), 65 Formula defining, 104 Frobenius Lemma, 229 Function Algebraic, 165 Hilbert, 92 Fundamental Theorem of Calculus, 170, 202 Gauss elimination, 66 Lemma, 45 Generalised Polynomial, 186 Gianni–Kalkbrener algorithm, 86 theorem, 86 Good reduction, 117, 127, 143 Graeffe method, 212 Greatest common divisor, 25, 43 domain, 43 Gröbner base, 74, 94 completely reduced, 75 Gröbner trace idea, 143 Hadamard bound, 208 Hensel Lifting Hybrid, 160 Linear, 155–156 Quadratic, 158 Hermite Algorithm, 172 Hilbert function, 92 polynomial, 93 History of Computer Algebra, 13 Ideal, 23 elimination, 87 polynomial, 72 Principal, 24 saturated, 99 Identity Bezout’s, 50 Inequality Landau, 210 Landau–Mignotte, 115 Initial, 95 Integer Algebraic, 165 Integral domain, 24 Integration indefinite, 170 IntExp–Polynomial Algorithm, 187 IntExp–Rational Expression Algorithm, 188 IntLog–Polynomial Algorithm, 184 IntLog–Rational Expression Algorithm, 185 Inverse matrix, 66 Knuth bound, 212 Landau Inequality, 210 Landau notation, 26 Landau–Mignotte Inequality, 115 Laurent
INDEX 265 Polynomial, 186 Least common multiple, 43 Lemma Frobenius, 229 Liouville’s Principle, 179 Parallel Version, 193 Macsyma, 232 Mahler bound, 212 measure, 209, 210 Main variable, 95 Maple, 234 Matrix Banded, 65 Circulant, 65 inverse, 66 Sylvester, 205 Toeplitz, 65 measure Mahler, 209, 210 Minimal Polynomial, 167 Monic polynomial, 31 Monomial, 36 leading, 72 Multiplicity of a solution, 56 MuPAD, 238 Newton series, 40 Noetherian ring, 24 Normal representation, 17 Normal Selection Strategy, 81 Number Algebraic, 165 Numerator, 41 Ordering admissible, 36 elimination, 78 matrix, 79 purely lexicographic, 78 total degree, then lexicographic, 78 total degree, then reverse lexicographic, 78 weighted, 79 Ostrogradski–Horowitz Algorithm, 173 Parallel Risch Algorithm, 194 Partial Cylindrical algebraic decomposition, 111 Polynomial definition, 29 factored representation, 34 generalised, 186 height of, 209 Hilbert, 93 Laurent, 186 length of, 209 minimal, 167 remainder sequence, 47 signed, 47 Wilkinson, 211 Positivstellensatz, 110 Prenex normal form, 103 Primitive (of polynomials), 45 p.r.s., 47 part, 45 Principal ideal, 24 domain, 24 Principle Liouville’s, 179 (Parallel Version), 193 Tarski–Seidenberg, 103 Problem Elementary integration, 181 Elementary Risch differential equation, 181 Projection, 109 Proposition algebraic, 101 semi-algebraic, 101 Pseudo-euclidean algorithm, 47 Pseudo-remainder, 47, 97 Quantifier alternation, 103 elimination, 103
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Contents 1 Introduction 13 1.1 Hist
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CONTENTS 3 4 Modular Methods 113 4.
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CONTENTS 5 A Algebraic Background 2
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List of Figures 2.1 A polynomial SL
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LIST OF FIGURES 9 List of Open Prob
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Preface This text is under active d
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Chapter 1 Introduction Computer alg
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1.1. HISTORY AND SYSTEMS 15 1.1 His
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1.2. EXPANSION AND SIMPLIFICATION 1
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1.3. ALGEBRAIC DEFINITIONS 23 which
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1.3. ALGEBRAIC DEFINITIONS 25 Propo
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1.5. SOME MAPLE 27 Definition 18 (F
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Chapter 2 Polynomials Polynomials a
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2.1. WHAT ARE POLYNOMIALS? 31 2.1.1
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2.1. WHAT ARE POLYNOMIALS? 33 MULTI
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2.1. WHAT ARE POLYNOMIALS? 35 not n
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2.1. WHAT ARE POLYNOMIALS? 37 While
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2.1. WHAT ARE POLYNOMIALS? 39 p:=x+
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2.2. RATIONAL FUNCTIONS 41 Proposit
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2.3. GREATEST COMMON DIVISORS 43 2.
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2.3. GREATEST COMMON DIVISORS 45 Le
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2.3. GREATEST COMMON DIVISORS 47 93
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2.3. GREATEST COMMON DIVISORS 49 a
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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
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: INDEX 263 Budan-Fourier theorem, 63
Page 267 and 268: INDEX 267 Toeplitz Matrix, 65 Trage
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