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82 CHAPTER 3. POLYNOMIAL EQUATIONS Definition 47 A polynomial is said to be homogeneous if every term has the same total degree. A set of polynomials is said to be homogeneous if each of them separately is homogeneous. Note that we are not insisting that all terms in the set have the same degree, merely that within each polynomial they have the same total degree. Definition 48 If f = ∑ ∏ n c i j=1 xai,j j ∈ K[x 1 , . . . , x n ] is not homogeneous, and has total degree d, we can define its homogenisation to be f 0 ∈ K[x 0 , x 1 , . . . , x n ] as f 0 = ∑ c i x d−∑ n j=1 ai,j 0 n∏ j=1 x ai,j j . Proposition 35 If f and g are homogeneous, so is S(f, g) and h where f → g h. Corollary 7 If the input to Algorithm 8 is a set of homogeneous polynomials, then the entire computation is carried out with homogeneous polynomials. The normal selection strategy is observed to work well with homogeneous polynomials, but can sometimes be very poor on non-homogeneous polynomials. Hence [GMN + 91] introduced the following concept. Definition 49 The ‘sugar’ S f of a polynomial f in Algorithm 8 is defined inductively as follows: 1. For an input f ∈ G 0 , S f is the total degree of f (even if we are working in a lexicographic ordering) 2. If t is a term, S tf = deg(t) + S f ; 3. S f+g = max(S f , S g ). We define the sugar of a pair of polynomials to be the sugar of their S-polynomial, i.e. (the notation is not optimal here!) S (f,g) = S S(f,g) . The sugar of a polynomial is then the degree it would have had we homogenised all the polynomials before starting Algorithm 8. Definition 50 We say that an implementation of Algorithm 8 follows a sugar selection strategy if, at each stage, we pick a pair (i, j) such that S (gi,g j) is minimal. This does not completely specify what to do, and it is usual to break ties with the normal selection strategy (Definition 46), and “sugar then normal” is generally just referred to as “sugar”.
3.3. NONLINEAR MULTIVARIATE EQUATIONS: DISTRIBUTED 83 3.3.5 A Matrix Formulation Equation (3.13) showed how a family of linear equations can be represented as a matrix equation. We can do the same with nonlinear equations: (3.22) can be written as ⎛ ⎞ ( 1 0 0 0 ) −1 0 1 0 0 −1 ⎜ ⎝ x 2 xy x y 1 ⎟ ⎠ = 0 (3.26) However, this does not give us an obvious solution. Rather, we need to extend the system, allowing not just the original equations, but also y times the first and x times the second, to give the following. Elimination in this gives us ⎛ x 2 y ⎛ ⎞ 1 0 0 0 −1 0 x 2 ⎜ 0 1 0 0 0 −1 ⎟ xy ⎝ ⎠ 1 0 0 −1 0 0 ⎜ x ⎝ 0 0 1 0 0 −1 y 1 ⎛ x 2 y ⎛ ⎞ 1 0 0 0 −1 0 x 2 ⎜ 0 1 0 0 0 −1 ⎟ xy ⎝ ⎠ 0 0 0 −1 1 0 ⎜ x ⎝ 0 0 1 0 0 −1 y 1 ⎞ = 0. (3.27) ⎟ ⎠ ⎞ = 0, (3.28) ⎟ ⎠ which produces, as the third row, the equation y − x, as we do (up to a change of sign) after (3.22). In pure linear algebra, we can do no further, since we really require y times this equation. This means considering ⎛ x 2 y 2 ⎛ ⎞ x 1 0 0 0 0 0 −1 0 0 2 y x 0 1 0 0 0 0 0 −1 0 2 xy 0 0 1 0 0 0 0 0 −1 2 xy ⎜ 1 0 0 0 −1 0 0 0 0 ⎟ ⎝ ⎠ x 0 1 0 0 0 −1 0 0 0 ⎜ y 0 0 0 0 1 0 0 0 −1 2 ⎝ y 1 ⎞ = 0. (3.29) ⎟ ⎠
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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
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
Page 65 and 66: 3.2. LINEAR EQUATIONS IN SEVERAL VA
Page 71 and 72: 3.3. NONLINEAR MULTIVARIATE EQUATIO
Page 81: 3.3. NONLINEAR MULTIVARIATE EQUATIO
Page 101 and 102: 3.5. EQUATIONS AND INEQUALITIES 101
Page 111 and 112: 3.6. CONCLUSIONS 111 Partial Proof.
Page 113 and 114: Chapter 4 Modular Methods In chapte
Page 115 and 116: 4.1. GCD IN ONE VARIABLE 115 The co
Page 117 and 118: 4.1. GCD IN ONE VARIABLE 117 This l
Page 119 and 120: 4.1. GCD IN ONE VARIABLE 119 be mad
Page 121 and 122: 4.1. GCD IN ONE VARIABLE 121 Figure
Page 123 and 124: 4.1. GCD IN ONE VARIABLE 123 Figure
Page 125 and 126: 4.2. POLYNOMIALS IN TWO VARIABLES 1
Page 131 and 132: 4.3. POLYNOMIALS IN SEVERAL VARIABL
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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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