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14 CHAPTER 1. INTRODUCTION Definition 1 A number n, or more generally any algebraic object, is said to be transcendental over a ring R if there is no non-zero polynomial p with coefficients in R such that p(n) = 0. With this definition, we can say that e π√ 163 is transcendental over the integers. We will see throughout this book (for an early example, see page 47) that innocent-seeming problems can give rise to numbers far greater than one would expect. Hence there is a requirement to deal with numbers larger, or to greater precision, than is provided by the hardware manufacturers whose chips underlie our computers. Practically every computer algebra system, therefore, has a package for manipulating arbitrary-sized integers (so-called bignums) or real numbers (bigfloats). These arbitrary-sized objects, and the fact that mathematical objects in general are unpredictably sized, means that computer algebra systems generally need sophistucated memory management, generally garbage collection (see [JHM11] for a general discussion of this topic). But the fact that the systems can deal with large numbers does not mean that we should let numbers increase without doing anything. If we have two numbers with n digits, adding them requires a time proportional to n, or in more formal language (see section 1.4) a time O(n). Multiplying them 4 requires a time O(n 2 ). Calculating a g.c.d., which is fundamental in the calculation of rational numbers, requires O(n 3 ), or O(n 2 ) with a bit of care 5 . This implies that if the numbers become 10 times longer, the time is multiplied by 10, or by 100, or by 1000. So it is always worth reducing the size of these integers. We will see later (an early example is on page 113) that much ingenuity has been wellspent in devising algorithms to compute “obvious” quantities by “non-obvious” ways which avoid, or reduce, the use of large numbers. The phrase intermediate expression swell is used to denote the phenomenon where intermediate quantites are much larger than the input to, or outputs from, a calculation. Notation 1 We write algorithms in an Algol-like notation, with the Rutishauser symbol := to indicate assignment, and = (as opposed to C’s ==) to indicate the equality predicate. We use indentation to indicate grouping 6 , rather than clutter the text with begin . . . end. Comments are introduced by the # character, running to the end of the line. 4 In principle, O(n log n log log n) is enough [AHU74, Chapter 8], but no computer algebra system routinely uses this, for it is more like 20n log n log log n. However, most systems will use ‘Karatsuba arithmetic’ (see [KO63, and section B.3]), which takes O(n log 2 3 ≈ n 1.585 ), once the numbers reach an appropriate length, often 16 words [SGV94]. 5 In principle, O(n log 2 n log log n) [AHU74, Chapter 8], but again no system uses it routinely, but this combined with Karatsuba gives O(n log 2 3 log n) (section B.3.6), and this is commonly used. 6 An idea which was tried in Axiom [JS92], but which turns out to be better in books than in real life.
1.1. HISTORY AND SYSTEMS 15 1.1 History and Systems The first recorded use of computers to do computations of the sort we envisage was in 1951 [MW51], where 180 ( 2 127 − 1 ) 2 − 1, a 79-digit number, was shown to be prime. In 1952 the great mathematician Emil Artin had the equally great computer scientist John von Neumann perform an extensive calculation relating to elliptic curves on the MANIAC computer [ST92, p. 119]. In 1953, two theses [Kah53, Nol53] kicked off the ‘calculus’ side of computer algebra with programs to differentiate expressions. In the same year, [Has53] showed that algorithms in group theory could be implemented on computers. 1.1.1 The ‘calculus’ side The initial work [Kah53, Nol53] consisted of programs to do one thing, but the focus soon moved on to ‘systems’, capable of doing a variety of tasks. One early one was Collins’ system SAC [Col71], written in Fortran. Its descendants continue in SAC-2 [Col85] and QEPCAD [Bro03]. Of course, computer algebra systems can be written in any computer language, even Cobol [fHN76]. However, many of the early systems were written in LISP, largely because of its support for garbage collection and large integers. The group at M.I.T., very influential in the 1960s and 70s, developed MACSYMA [MF71, PW85]. This system now exists in several versions [Ano07]. At about the same time, Hearn developed Reduce [Hea05], and shortly after a group at IBM Yorktown Heights produced SCRATCHPAD [GJY75]. This group then produced AXIOM [JS92], a system that attempted to match the generality of mathematics with ‘generic programming’ to allow algorithms to be programmed in the generality in which they are conventionally (as in this book) stated. These were, on the whole, very large software systems, and attempts were made to produce smaller ones. muMATH [RS79] and its successor DERIVE [RS92] were extremely successful systems on the early PC, and paved the way for the computer algebra facilities of many high-end calculators. Maple [CGGG83] pioneered a ‘kernel+library’ design. The basic Maple system, the kernel, is a relatively small collection of compiled C code. When a Maple session is started, the entire kernel is loaded. It contains the essential facilities required to run Maple and perform basic mathematical operations. These components include the Maple programming language interpreter, arithmetic and simplification routines, print routines, memory management facilities, and a collection of fundamental functions. [MGH + 03, p. 6] 1.1.2 The ‘group theory’ side Meanwhile, those interested in computation group theory, and related topics, had not been idle. One major system developed during the 1970s/80s was CAY- LEY [BC90]. This team later looked at Axiom, and built the system MAGMA
Page 1 and 2: Contents 1 Introduction 13 1.1 Hist
Page 3 and 4: CONTENTS 3 4 Modular Methods 113 4.
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: Chapter 1 Introduction Computer alg
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
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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
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A.2. USEFUL ESTIMATES 213 centring
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A.3. CHINESE REMAINDER THEOREM 215
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A.5. VANDERMONDE SYSTEMS 217 Clearl
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A.5. VANDERMONDE SYSTEMS 219 wherea
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Appendix B Excursus This appendix i
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B.2. EQUALITY OF FACTORED POLYNOMIA
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B.3. KARATSUBA’S METHOD 225 addit
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B.4. STRASSEN’S METHOD 227 answer
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B.4. STRASSEN’S METHOD 229 If the
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Appendix C Systems This appendix di
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C.2. MACSYMA 233 (1) -> a:=1 Figure
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C.3. MAPLE 235 > (x^2-1)^10/(x+1)^1
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C.3. MAPLE 237 Table C.2: Another s
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C.5. REDUCE 239 1: (x^2-1)^10/(x+1)
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Appendix D Index of Notation Notati
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Bibliography [Abb02] J.A. Abbott. S
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BIBLIOGRAPHY 245 [BCM94] [BCR98] [B
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BIBLIOGRAPHY 247 [Buc79] [Buc84] [B
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BIBLIOGRAPHY 249 [CW90] D. Coppersm
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BIBLIOGRAPHY 251 [FGT01] E. Fortuna
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BIBLIOGRAPHY 253 [Isa85] I.M. Isaac
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BIBLIOGRAPHY 255 [Loo82] R. Loos. G
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BIBLIOGRAPHY 257 [Per09] [PQR09] [P
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BIBLIOGRAPHY 259 [SS11] J. Schicho
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BIBLIOGRAPHY 261 [Zip79b] R.E. Zipp
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INDEX 263 Budan-Fourier theorem, 63
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INDEX 265 Polynomial, 186 Least com
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INDEX 267 Toeplitz Matrix, 65 Trage
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