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54 CHAPTER 2. POLYNOMIALS 5 ′ x ∗ y = y ∗ x; where x and y are indeterminates, not general polynomials. This means that some of the traditional laws of algebra cease to operate: for example becomes (ax + b)(ax − b) = a 2 x 2 − b 2 (ax + b)(ax − b) = a 2 x 2 + (−a ∗ b + b ∗ a)x − b 2 2. The coefficients commute, but the indeterminates do not. For definiteness, we will refer to this case as polynomials with non-commuting indeterminates. In this case, rule 5 of definition 19 has to be replaced by the assumption • m ⊗ n = n ⊗ m. At this point, many of the traditional laws of algebra cease to operate: even the Binomial Theorem in the form has to be replaced by (x + y) 2 = x 2 + 2xy + y 2 (x + y) 2 = x 2 + (xy + yx) + y 2 . A common case of non-commuting indeterminates is in differential algebra, where the variable x and the differentiation operator d dx do not commute, but rather satisfy the equation d da (xa) = x + a. (2.12) dx dx 3. Neither can be assumed to commute, in which case rule 5 of definition 19 is just deleted, with no replacement. Notation 14 If the variables do not commute, it is usual to use the notation R〈x 1 , . . . , x n 〉 for the ring of polynomials with coefficients in R and the noncommuting variables x 1 , . . . , x n .
Chapter 3 Polynomial Equations In the first parts of this chapter, we will deal with polynomial equations, either singly or as sets of equations. A preliminary remark is in order. Any polynomial equation A = B, (3.1) where A and B are polynomial equations, can be reduced to one whose righthand side is zero, i.e. A − B = 0. (3.2) Notation 15 Henceforth, all polynomial equations will be assumed to be in the form of (3.2). 3.1 Equations in One Variable We may as well assume that the unknown variable is x. If the equation is linear in x then, by the notation above, it takes the form The solution is then obvious: x = −b/a. 3.1.1 Quadratic Equations ax + b = 0. (3.3) Again, by the notation above, our equation takes the form ax 2 + bx + c = 0. (3.4) The solutions are well-known to most schoolchildren: there are two of them, of the form x = −b ± √ b 2 − 4ac . (3.5) 2a 55
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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 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: 2.4. NON-COMMUTATIVE POLYNOMIALS 53
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
Page 71 and 72: 3.3. NONLINEAR MULTIVARIATE EQUATIO
Page 101 and 102: 3.5. EQUATIONS AND INEQUALITIES 101
Page 103 and 104: 3.5. EQUATIONS AND INEQUALITIES 103
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3.5. EQUATIONS AND INEQUALITIES 105
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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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