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162 CHAPTER 5. P -ADIC METHODS Figure 5.8: Overview of Factoring Algorithm Algorithm 30 (Factor over Z) Input: A primitive square-free f(x) ∈ Z[x] Output: The factorization of f into irreducible polynomials over the integers. p 1 := find_prime(f); F 1 :=Proposition 51(f, p 1 ) if F 1 is a singleton return f S :=AllowableDegrees(F 1 ) for i := 2, . . . , 5 #5 from [Mus78] p i := find_prime(f); F i :=Proposition 51(f, p i ) S := S∩AllowableDegrees(F i ) if S = ∅ return f p :=best({p i }); F := F i ; #‘best’ in terms of fewest factors F :=Algorithm 26(f, F, p, log p (2LM(f))) return Algorithm 29(p log p (2LM(f)) , f, F ) find_prime(f) returns a prime p such that f remains square-free modulo p. AllowableDegrees(F ) returns the set of allowable proper factor degrees from a modular factorization. Algorithm 29 can be replaced by any of improvements 1–5 on pages 161–161.
5.8. MULTIVARIATE FACTORING 163 Open Problem 12 (Better Choice of ‘Best’ Prime) In algorithm 30, we picked the ‘best’ prime and the corresponding factorization. In principle, it is possible to do better, as in the following example, but the author has seen no systematic treatment of this. Example 10 Suppose f p factors as polynomials of degree 2, 5 and 5, and f q factors as polynomials of degree 3, 4 and 5. The factorization modulo p implies that the allowable degrees of (proper) factors are 2, 5, 7 and 10. The factorization modulo q implies that the allowable degrees of (proper) factors are 3, 4, 5, 7, 8 and 9. Hence the only possible degrees of proper factors over the integers are 7 and 12. The factorization modulo p can yield this in two ways, but the fatcorization modulo q can only yield this by combining the factors of degrees 3 and 4. Hence we should do this, and lift the factorization of f q . Open Problem 13 (Low-degree Factorization) Although no system to the author’s knowledge implements it, it would be possible to implement an efficient procedure to find all factors of limited total degree d. This would be efficient for three reasons: • the Cantor–Zassenhaus algorithm (section 5.3.3) need only run up to maximum degree d; • it should be possible to use smaller bounds on the lifting • The potentially-exponential recombination process need only run up to total degree d. In particular, if d = 1, no recombination is needed. At the moment, the user who only wants low-degree factors has to either program such a search, or rely on the system’s factor, which may waste large amounts of time looking for high-degree factors, which the user does not want. 5.8 Multivariate Factoring 5.9 Other Applications 5.10 Conclusions
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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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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
Page 139 and 140: 4.4. FURTHER APPLICATIONS 139 2 7 3
Page 141 and 142: 4.4. FURTHER APPLICATIONS 141 i.e.
Page 143 and 144: 4.5. GRÖBNER BASES 143 Observation
Page 145 and 146: 4.5. GRÖBNER BASES 145 Table 4.2:
Page 147 and 148: 4.5. GRÖBNER BASES 147 Figure 4.17
Page 149 and 150: Chapter 5 p-adic Methods In this ch
Page 151 and 152: 5.2. MODULAR METHODS 151 nomials, b
Page 153 and 154: 5.4. FROM Z P TO Z? 153 Figure 5.3:
Page 155 and 156: 5.5. HENSEL LIFTING 155 polynomials
Page 157 and 158: 5.5. HENSEL LIFTING 157 Figure 5.4:
Page 159 and 160: 5.5. HENSEL LIFTING 159 Figure 5.7:
Page 161: 5.7. UNIVARIATE FACTORING SOLVED 16
Page 165 and 166: Chapter 6 Algebraic Numbers and fun
Page 167 and 168: 6.1. THE D5 APPROACH TO ALGEBRAIC N
Page 169 and 170: Chapter 7 Calculus Throughout this
Page 171 and 172: 7.2. INTEGRATION OF RATIONAL EXPRES
Page 177 and 178: 7.3. THEORY: LIOUVILLE’S THEOREM
Page 183 and 184: 7.4. INTEGRATION OF LOGARITHMIC EXP
Page 185 and 186: 7.5. INTEGRATION OF EXPONENTIAL EXP
Page 191 and 192: 7.7. THE RISCH DIFFERENTIAL EQUATIO
Page 193 and 194: 7.8. THE PARALLEL APPROACH 193 i.e.
Page 195 and 196: 7.10. OTHER CALCULUS PROBLEMS 195 7
Page 197 and 198: Chapter 8 Algebra versus Analysis W
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
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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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