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Index Abel’s theorem, 58 Additive complexity, 39 Admissible orderings, 36 Algebraic closure, 25 curve, 61 decomposition, 104 block-cylindrical, 108 cylindrical, 106 sampled, 105 function, 165 integer, 165 number, 165 proposition, 101 variable, 95 Algorithm Bareiss, 69 Buchberger, 75 Cantor–Zassenhaus, 152 Chinese Remainder, 215 for Polynomials, 216 Multivariate, 217 Polynomial form, 216 Euclid’s, 44 Extended Euclid, 49 Extended Subresultant, 50 Faugère–Gianni–Lazard–Mora, 88 Hermite’s, 172 IntExp–Polynomial, 187 IntExp–Rational Expression, 188 IntLog–Polynomial, 184 IntLog–Rational Expression, 185 Ostrogradski–Horowitz, 173 Parallel Risch, 194 Primitive p.r.s., 47 Sturm sequence evaluation, 63 Subresultant, 48 Trager–Rothstein, 175 Vandermonde solver, 218 variant, 219 Alternations of quantifiers, 103 Ascending Chain Condition, 24 Associates, 24 Associativity, 23 Assumption Zippel, 134 Axiom, 231 Bad reduction, 117, 127, 143 Banded Matrix, 65 Bareiss algorithm, 69 Basis completely reduced, 75 Gröbner, 74, 94 completely reduced, 75 shape, 91 Bezout’s Identity, 50 Birational equivalence, 61 Block-cylindrical decomposition, 108 Bound, 208 Cauchy, 211–212 Hadamard, 208 Knuth, 212 Mahler, 212 Buchberger Algorithm, 75 Criterion First, 81 gcd, 81 lcm, 81 Third, 81 Theorem, 75 262
INDEX 263 Budan–Fourier theorem, 63, 222 Calculus Fundamental Theorem of, 170, 202 Candid representation, 17 Canonical representation, 17 Cantor–Zassenhaus Algorithm, 152 Cauchy bound, 211 Cell, 104 Chain Regular, 96 Chain Condition Ascending, 24 Descending, 36 Characteristic, 25 set, 95 Chebotarev Density Theorem, 154 Chinese Remainder Theorem, 215 (Polynomial), 216 Circulant Matrix, 65 Closure Zariski, 97 Coefficient leading, 31 of a polynomial, 29 Commutativity, 23 Complexity additive, 39 Constant definition of, 170 Content (of polynomials), 45 Cylinder, 106 Cylindrical algebraic decomposition, 106 Partial, 111 Decomposition algebraic, 104 block-cylindrical, 108 cylindrical, 106 partial, 111 equiprojectable, 96 Lemma (exponential), 186 Lemma (logarithmic), 182 Lemma (rational expressions), 171 sign-invariant, 106 square-free, 56 Defining formula, 104 Degree of polynomial, 31 Denominator, 41 common, 41 Dense matrix, 65 polynomial, 31 Descartes rule of signs, 63, 221 Descending Chain Condition, 36 Difference field, 195 ring, 195 Differential field, 169 ring, 169 Dimension ideal, 77 linear space, 71 mixed, 77 triangular set, 95 Directed Acyclic Graph, 38 Discriminant, 207 Distributed representation, 36 Distributivity, 23 Division, 30 Dodgson–Bareiss theorem, 68 Elementary expression, 178 generator, 176 Elimination Gauss, 66 fraction-free, 69 ideal, 87 ordering, 78 Equality algebraic, 42 Equiprojectable decomposition, 96 variety, 96 Equivalent, 72 Euclid’s Algorithm, 44 Algorithm (Extended), 49, 141 Theorem, 44
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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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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
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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: BIBLIOGRAPHY 261 [Zip79b] R.E. Zipp
Page 265 and 266: INDEX 265 Polynomial, 186 Least com
Page 267 and 268: INDEX 267 Toeplitz Matrix, 65 Trage
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