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242 APPENDIX D. INDEX OF NOTATION Notation Meaning Reference Page M(f) The Mahler measure of f(x) Notation 28 209 M(f) The Mahler measure of f(x 1 , . . . , x n ) Definition 91 210 M(n) The time needed to multiply two objects of length n Notation 31 227 mon(H) The (multiplicative) monoid of H Theorem 26 110 N The nonnegative integers (natural numbers) Newton(p) The Newton series Notation 13 40 num(f) The numerator of f Proposition 10 41 Q The rational numbers R The real numbers S f The “sugar’ of the polynomial f Definition 49 82 S (f,g) The “sugar’ of the polynomial pair (f, g) Definition 49 82 Sqrt The multivalued square root Notation 26 200 S(f, g) The S-polynomial of f and g Definition 42 73 Sk(f) The skeleton of the polynomial f Definition 72 134 Z The integers The integers modulo N Z N
Bibliography [Abb02] J.A. Abbott. Sparse Squares of Polynomials. Math. Comp., 71:407–413, 2002. [Abb04] J.A. Abbott. CoCoA: a laboratory for computations in commutative algebra. ACM SIGSAM Bulletin 1, 38:18–19, 2004. [Abb09] J.A. Abbott. Bounds on Factors in Z[x]. http://arxiv.org/ abs/0904.3057, 2009. [ABD85] [ABD88] [Ada43] [AHU74] [AHU83] [Akr82] [AL94] [ALMM99] J.A. Abbott, R.J. Bradford, and J.H. Davenport. A Remark on Factorisation. SIGSAM Bulletin 2, 19:31–33, 1985. J.A. Abbott, R.J. Bradford, and J.H. Davenport. Factorisation of Polynomials: Old Ideas and Recent Results. In R. Janssen, editor, Proceedings “Trends in Computer Algebra”, pages 81–91, 1988. Ada Augusta Countess of Lovelace. Sketch of the Analytical Engine invented by Charles Babbage, by L.F. Menabrea of Turin, with notes on the memoir by the translator. Taylor’s Scientific Memoirs 29, 3:666–731, 1843. A.V. Aho, J.E. Hopcroft, and J.D. Ullman. The Design and Analysis of Computer Algorithms. Addison-Wesley, 1974. A.V. Aho, J.E. Hopcroft, and J.D. Ullman. Data Structures and Algorithms. Addison-Wesley, 1983. A.G. Akritas. Reflections on a Pair of Theorems by Budan and Fourier. Mathematics Magazine, 55:292–298, 1982. W.W. Adams and P. Loustaunau. An introduction to Gröbner bases. Amer. Math. Soc., 1994. P. Aubry, D. Lazard, and M. Moreno Maza. On the Theories of Triangular Sets. J. Symbolic Comp., 28:105–124, 1999. [ALSU07] A.V. Aho, M.S. Lam, R. Sethi, and J.D. Ullman. Compilers: Principles, Techniques and Tools. Pearson Addison-Wesley, 2007. 243
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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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Page 191 and 192: 7.7. THE RISCH DIFFERENTIAL EQUATIO
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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: Appendix D Index of Notation Notati
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 Budan-Fourier theorem, 63
Page 265 and 266: INDEX 265 Polynomial, 186 Least com
Page 267 and 268: INDEX 267 Toeplitz Matrix, 65 Trage
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