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246 BIBLIOGRAPHY [BHPR11] [BL98] [BMMT94] O. Bastani, C.J. Hillar, P. Popov, and J.M. Rojas. Randomization, Sums of Squares, and Faster Real Root Counting for Tetranomials and Beyond. http://arxiv.org/abs/1101.2642, 2011. T. Breuer and S.A. Linton. The GAP4 Type System Organising Algebraic Algorithms. In O.Gloor, editor, Proceedings ISSAC ’98, pages 38–45, 1998. E. Becker, M.G. Marinari, T. Mora, and C. Traverso. The shape of the shape lemma. In Proceedings ISSAC 1994, pages 129–133, 1994. [Bou61] N. Bourbaki. Algèbre Commutative, chapter 2. Hermann, 1961. [Bou70] N. Bourbaki. Théorie des Ensembles. Diffusion C.C.L.S., 1970. [BPR06] [Bro69] S. Basu, R. Pollack, and M.-F. Roy. Algorithms in Real Algebraic Geometry, 2nd ed. Springer, 2006. W.S. Brown. Rational Exponential Expressions, and a conjecture concerning π and e. Amer. Math. Monthly, 76:28–34, 1969. [Bro71a] W.S. Brown. On Euclid’s Algorithm and the Computation of Polynomial Greatest Common Divisors. In Proceedings SYMSAC 1971, pages 195–211, 1971. [Bro71b] W.S. Brown. On Euclid’s Algorithm and the Computation of Polynomial Greatest Common Divisors. J. ACM, 18:478–504, 1971. [Bro90] M. Bronstein. Integration of elementary function. J. Symbolic Comp., 9:117–173, 1990. [Bro91] [Bro03] [Bro07] [BS86] [Buc70] M. Bronstein. The Algebraic Risch Differential Equation. In Proceedings ISSAC 91, pages 241–246, 1991. C.W. Brown. QEPCAD B: a program for computing with semialgebraic sets using CADs. ACM SIGSAM Bulletin 4, 37:97–108, 2003. M. Bronstein. Structure theorems for parallel integration (Paper completed by Manuel Kauers). J. Symbolic Comp., 42:757–769, 2007. D. Bayer and M. Stillman. The Design of Macaulay: A System for Computing in Algebraic Geometry and Commutative Algebra. In Proceedings SYMSAC 86, pages 157–162, 1986. B. Buchberger. Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystem (English translation in [Buc98]). Aequationes Mathematicae, 4:374–383, 1970.
BIBLIOGRAPHY 247 [Buc79] [Buc84] [Buc98] [BW93] [CA76] B. Buchberger. A Criterion for Detecting Unnecessary Reductions in the Construction of Groebner Bases. In Proceedings EUROSAM 79, pages 3–21, 1979. B. Buchberger. A critical pair/completion algorithm for finitely generated ideals in rings. Logic and Machines: Decision Problems and Complexity, pages 137–155, 1984. B. Buchberger. An Algorithmic Criterion for the Solvability of a System of Algebraic Equations. In Gröbner Bases and Applications, pages 535–545, 1998. T. Becker and V. (with H. Kredel) Weispfenning. Groebner Bases. A Computational Approach to Commutative Algebra. Springer Verlag, 1993. G.E. Collins and A.V. Akritas. Polynomial Real Root Isolation Using Descartes’ Rule of Signs. In R.D. Jenks, editor, Proceedings SYMSAC 76, pages 272–275, 1976. [Car58] C. Carathéodory. Theory of functions of a complex variable. Chelsea Publ., 1958. [Car73] [Car04] [Cau29] [CD85] [CD91] [CDJW00] [CFG + 84] [CGG84] H. Cartan. Elementary Theory of Analytic Functions of One or Several Complex Variables. Addison-Wesley, 1973. J. Carette. Understanding Expression Simplification. In J. Gutierrez, editor, Proceedings ISSAC 2004, pages 72–79, 2004. A.-L. Cauchy. Exercices de Mathématiques Quatrième Année. De Bure Frères, 1829. D. Coppersmith and J.H. Davenport. An Application of Factoring. J. Symbolic Comp., 1:241–243, 1985. D. Coppersmith and J.H. Davenport. Polynomials whose Powers are Sparse. Acta Arithmetica, 58:79–87, 1991. R.M. Corless, J.H. Davenport, D.J. Jeffrey, and S.M. Watt. According to Abramowitz and Stegun, or arccoth needn’t be uncouth. SIGSAM Bulletin 2, 34:58–65, 2000. B.W. Char, G.J. Fee, K.O. Geddes, G.H. Gonnet, M.B. Monagan, and S.M. Watt. On the Design and Performance of the Maple System. Technical Report CS-84-13, 1984. B.W. Char, K.O. Geddes, and G.H. Gonnet. GCDHEU: Heuristic Polynomial GCD Algorithm Based on Integer GCD Computation. In J.P. Fitch, editor, Proceedings EUROSAM 84, pages 285–296, 1984.
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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.
Page 195 and 196: 7.10. OTHER CALCULUS PROBLEMS 195 7
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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: BIBLIOGRAPHY 245 [BCM94] [BCR98] [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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