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90 CHAPTER 3. POLYNOMIAL EQUATIONS 1 (irreducble) c (irreducble) c 2 c 3 (irreducble) (irreducble) c 4 which reduces to − 185 14 − 293 42 a3 − 1153 42 a2 b + 509 7 ab2 − 323 821 173 21 abc + 6 b2 c − 751 14 ac2 + 626 21 bc2 + 31 42 c3 − 449 550 429 184 21 ac − 7 bc + 21 c2 − 407 6 a − 281 42 b − 4799 42 c . 42 b3 − 2035 42 a2 c − 14 a2 + 1165 772 14 ab − 21 b2 + c 20 which reduces to − 156473200555876438 7 + 1355257348062243268 21 bc 2 − 2435043982608847426 21 a 2 c − 455474473888607327 3 a − 87303768951017165 21 b − 5210093087753678597 21 c + 1264966801336921700 7 ab − 995977348285835822 7 bc − 2106129034377806827 21 abc + 136959771343895855 3 b 2 c + 1119856342658748374 21 ac + 629351724586787780 21 c 2 − 774120922299216564 7 ac 2 − 1416003666295496227 21 a 2 b + 1196637352769448957 7 ab 2 − 706526575918247673 7 a 2 − 1536916645521260147 21 b 2 − 417871285415094524 21 a 3 − 356286659366988974 21 b 3 + 373819527547752163 21 c 3 , which can be expressed in terms of the previous ones as p = −1 + 6 c + 41 c 2 − 71 c 3 + 41 c 18 − 197 c 14 − 106 c 16 + 6 c 19 − 106 c 4 − 71 c 17 − 92 c 5 − 197 c 6 − 145 c 7 − 257 c 8 − 278 c 9 − 201 c 10 − 278 c 11 − 257 c 12 − 145 c 13 − 92 c 15 . The polynomial c 20 − p gets added to H: all higher powers of c are therefore expressible, and need not be enumerated. b which can be expressed in terms of the previous ones as q = − 9741532 1645371 − 8270 343 c + 32325724 548457 c2 + 140671876 1645371 c3 − 2335702 548457 c18 + 13420192 182819 c14 + 79900378 1645371 c16 + 1184459 1645371 c19 + 3378002 42189 c4 − 5460230 182819 c17 + 688291 4459 c5 + 1389370 11193 c6 + 337505020 1645371 c7 + 118784873 548457 c 8 + 271667666 1645371 c9 + 358660781 1645371 c10 + 35978916 182819 c11 + 193381378 1645371 c12 + 553986 3731 c13 + 43953929 548457 c15 . b − q is added to H: and all multiples of b are therefore expressible, and need not be enumerated. a which can be expressed in terms of the previous ones as r = 487915 4406102 705159 c − 16292173 705159 c14 − 17206178 705159 c2 − 1276987 235053 c16 − 91729 705159 c18 − 705159 c19 + 377534 705159 − 801511 26117 c3 − 26686318 705159 c4 + 4114333 705159 c17 − 34893715 705159 c5 − 37340389 705159 c6 − 409930 6027 c7 − 6603890 100737 c8 − 14279770 235053 c9 − 15449995 235053 c10 − 5382578 100737 c11 − 722714 18081 c12 − 26536060 705159 c13 − 13243117 705159 c15 . a − r is added to H, and there are no more monomials to consider. These last three give us the Gröbner base in a purely lexicographical order, which looks like { c 20 + · · · , b + · · · , a + · · ·}. As there are twenty solutions in reasonably general position (the polynomial in c alone does factor, but is squarefree), we only need one polynomial per variable, as is often the case. The existence of this algorithm leads to the following process for ‘solving’ a set of polynomial equations.
3.3. NONLINEAR MULTIVARIATE EQUATIONS: DISTRIBUTED 91 Algorithm 12 Input: A set S of polynomials Output: A ‘description of solutions’ G :=Buchberger(S, > tdeg ) if G is not zero-dimensional (Proposition 32) then return “not zero-dimensional” else H :=FGLM(G, > plex ) Use Gianni–Kalkbrener to solve H Other ways of computing Gröbner bases will be looked at in section 4.5. 3.3.9 The Shape Lemma Let us look again at example 2 of section 3.3.7. Here we needed three equations to define an ideal in two variables. We note that interchanging the rôles of x and y does not help (in this case, it might in others). However, using other coordinates than x and y definitely does. If we write the equations in terms of u = x + y, v = x − y instead, we get the basis [−4 v + v 3 , v 2 − 4 + 2 u] : (3.32) three values for v (0, 2 and −2), each with one value of u (2, 0 and 0 respectively), from which the solutions in x, y can be read off. Note that ordering v before u would give the basis [u 2 − 2 u, uv, v 2 − 4 + 2 u], (3.33) which is not of this form: it has three polynomials in two variables. This kind of operation is called, for obvious reasons, a rotation. Almost all rotations will place the equations “in general position”: and many theoretical approaches to these problems assume a “generic rotation” has been performed. In practice, this is a disaster, since sparsity is lost. Definition 51 ([BMMT94]) A basis for a zero-dimensional ideal is a shape basis if it is of the form {g 1 (x 1 ), x 2 − g 2 (x 1 ), . . . , x n − g n (x 1 )} . This is a Gröbner basis for any ordering in which ‘degree in x 1 ’ is the first criterion: in the terminology of matrix orderings (page 79), any ordering where the first row of the matrix is (λ, 0 . . . , 0). For a shape basis, the Gianni–Kalkbrener process is particularly simple: “determine x 1 and the rest follows”. Almost all zero-dimensional ideals have shape bases. The precise criterion (•) in the theorem below is somewhat technical, but is satisfied if there are no repeated components. Theorem 19 (Shape lemma) [BMMT94, Corollary 3] After a generic rotation, a zero-dimensional ideal has a shape basis if, and only if,
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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
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 89: 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
Page 139 and 140: 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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