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22 CHAPTER 1. INTRODUCTION • The simplify command is used to apply simplification rules to an expression. • The simplify/expr calling sequence searches the expression, expr, for function calls, square roots, radicals, and powers. It then invokes the appropriate simplification procedures. • symbolic Specifies that formal symbolic manipulation of expressions is allowed without regard to the analytical issue of branches for multi-valued functions. For example, the expression sqrt(x^2) simplifies to x under the symbolic option. Without this option, the simplified result must take into account the different possible values of the (complex) sign of x. Maple does its best to return a normal form, but can be fooled: for example RootOf ( Z 4 + b Z 2 + d ) √ − 1/2 −2 b + 2 √ b 2 − 4 d, which is actually zero (applying figure 3.2), does not simplify to zero under Maple 11. Because simplification may often require expansion, e.g. to take (x−1)(x+1) to x 2 − 1, the two are often confused, and indeed both Macsyma and Reduce (internally) used ratsimp and *simp (respectively) to denote what we have called expansion. 1.2.4 An example of simplification This section is inspired by an example in [Sto12]. Consider (cos (x)) 3 sin (x) + 1 2 (cos (x))3 sin (x) + 2 (cos (x)) 3 cos (2 x) sin (x) + 1 2 (cos (x))3 cos (4 x) sin (x) − 3 2 cos (x) (sin (x))3 − 2 cos (x) cos (2 x) (sin (x)) 3 − 1 2 cos (x) cos (4 x) (sin (x))3 . (1.2) Typing this into Maple simply collects terms, giving 3 2 (cos (x))3 sin (x) + 2 (cos (x)) 3 cos (2 x) sin (x) + 1 2 (cos (x))3 cos (4 x) sin (x) − 3 2 cos (x) (sin (x))3 − 2 cos (x) cos (2 x) (sin (x)) 3 − 1 2 cos (x) cos (4 x) (sin (x))3 . (1.3) combine(%,trig), i.e. using the multiple angle formulae to replace trigonometric powers by sin / cos of multiples of the angles, gives 3 8 sin (4 x) + 1 4 sin (2 x) + 1 4 sin (6 x) + 1 16 sin (8 x) . (1.4) expand(%,trig) (i.e. using the multiple angle formulae in the other direction) gives 4 sin (x) (cos (x)) 7 − 4 (cos (x)) 5 (sin (x)) 3 , (1.5)
1.3. ALGEBRAIC DEFINITIONS 23 which is also given by Maple’s simplify applied to (1.3) (simplify applied to (1.4) leaves it unchanged). Mathematica’s FullSimplify gives 2 (sin(3x) − sin(x)) (cos(x)) 5 . (1.6) However, an even simpler form is found by Eureqa 10 : (cos(x)) 4 sin(4x). (1.7) We note that (1.4) and (1.5) are the results of algorithmic procedures, pushing identities in one direction or the other, applied to (1.3), while (1.6) and (1.7) are half-way positions, which happen to be shorter than the algorithmic results. 1.3 Algebraic Definitions In this section we give some classic definitions from algebra, which we will return to throughout this book. Other concepts are defined as they occur, but these ones are assumed. Definition 6 A set R is said to be a ring if it has two binary operations + and ∗, a unary operation − and a distinguished element 0, such that, for all a, b and c in R: 1. a + (b + c) = (a + b) + c (associativity of +); 2. a ∗ (b ∗ c) = (a ∗ b) ∗ c (associativity of ∗); 3. a + b = b + a (commutativity of +); 4. a + (−a) = 0; 5. a + 0 = a; 6. a ∗ (b + c) = (a ∗ b) + (a ∗ c) (distributivity of ∗ over +); 7. a ∗ b = b ∗ a (commutativity of ∗). Not every text includes the last clause, and they would call a ‘commutative ring’ what we have called simply a ‘ring’. In the absence of the last clause, we will refer to a ‘non-commutative ring’. Definition 7 If R is a (possibly non-commutative) ring and ∅ ̸= I ⊆ R, then we say that I is a (left-)ideal of R, written I ⊳ R, if the following two conditions are satisfied 11 : 10 http://creativemachines.cornell.edu/eureqa. However, we should note that Maple (resp. Mathematica) has proved that (1.5) (resp. (1.6)) is equivalent to (1.2), whereas Eureqa merely claims that (1.7) fits the same data points as [a finite sample of] (1.2). Nevertheless, Eureqa’s capability to find a short equivalent in essentially Carette’s sense [Car04] is impressive, and in any case, knowing what to prove, any algebra system can prove equivalence. 11 We write f − g ∈ I since then 0 = f − f ∈ I, and then f + g = f − (0 − g) ∈ I.
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 and 14: Chapter 1 Introduction Computer alg
Page 15 and 16: 1.1. HISTORY AND SYSTEMS 15 1.1 His
Page 17 and 18: 1.2. EXPANSION AND SIMPLIFICATION 1
Page 19 and 20: 1.2. EXPANSION AND SIMPLIFICATION 1
Page 21: 1.2. EXPANSION AND SIMPLIFICATION 2
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
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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
Page 65 and 66: 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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