Abel's theorem in problems and solutions - School of Mathematics

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252 Appendix by Khovanskii choice: the germs of the basic functions are the germs of the constant functions (at every point of the space the allowed operations are the arithmetic operations, the integration, and raising to the power of the integral. 2) The class of germs of functions in representable by generalized quadratures (over the field of the constants) is defined by the following choice: the germs of the basic functions are the germs of the constant functions (at every point of the space the allowed operations are the arithmetic operations, the integration, the raising to the power of the integral, and the solution of algebraic equations. Notice that the above definitions can be translated almost literally in the case of abstract differential fields, provided with commutative differentiation operations In such a generalized form these definitions are owed to Kolchin. Now consider the class of the germs of functions representable by quadratures and by generalized quadratures in the spaces of any dimension Repeating the Liouville argument (cf., Theorem 1 in §A.2), it is not difficult to prove that the class of the germs of functions of several variables representable by quadratures and by generalized quadratures contains the germs of the rational functions of several variables and the germs of all elementary basic functions; these classes of germs are closed with respect to the composition. (The closure with respect to the composition of a class of germs of functions representable by quadratures means the following: if are germs of functions representable by quadratures at a point and is a germ of a function representable by quadratures at the point where then the germ at the point is the germ of a function representable by quadratures). A.14 Does there exist a class of germs of functions of several variables sufficiently wide (containing the germs of functions representable by generalized quadratures, the germs of entire functions of several variables and closed with respect to the natural operations such as the composition) for which the monodromy group is defined? In this section we define the class of and we state the theorem about the closure of this class with respect to the natural operations: this gives an affirmative answer to
Solvability of Equations 253 the question posed. I discovered the class of relatively recently: up to that time I believed the answer were negative. In the case of functions of a single variable it was useful to introduce the class of the Let us start with a direct generalization of the class of to the multi-dimensional case. A subspace in a connected analytic manifold M is said to be thin if there exists a countable set of open subsets and a countable set of analytic subspaces in these open subsets such that An analytic multi-valued function on the manifold M is called an if the set of its singular points is thin. Let us make this definition more precise. Two regular germs and given at the points and of the manifold M, are said to be equivalent if the germ is obtained by a regular continuation of the germ along some curve. Every germ equivalent to the germ is also called a regular germ of the analytic multi-valued function generated by the germ A point is said to be singular for the germ if there exists a curve such that the germ cannot be regularly continued along this curve, but for every this germ can be continued along the shortened curve It is easy to see that the sets of singular points for equivalent germs coincide. A regular germ is said to be an if the set of its singular points is thin. An analytic multi-valued function is called an if every one of its regular germs is an REMARK. For functions of one complex variable we have already given two definitions of The first one is the above definition, the second one is given by the theorem in §A.5. These definitions obviously coincide. For of several variables the notions of monodromy group and of monodromy pair are automatically translated. Let us clarify way the multi-dimensional case is more complicated than the one-dimensional case. Imagine the following situation. Let be a multi-valued analytic function of two variables with a set A of branch points, where is an analytic curve on the complex plane. It can happen that at one of the points there exists an analytic germ of the multi-valued analytic function (by the definition of the set A of branch points, at the point not every germ of the function exists; yet some of them can exist). Now let and be two analytic functions of the complex variable
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ABEL’S THEOREM IN PROBLEMS AND SO
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Abel’s Theorem in Problems and So
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Contents Preface for the English ed
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A.10.3 The integrable case A.11 Top
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Preface to the English edition by V
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of the differential equations. Unfo
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Preface In high school algebraic eq
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Introduction We begin this book by
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Introduction 3 where by is indicate
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Introduction 5 By Viète’s theore
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Introduction 7 We will be able to p
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Chapter 1 Groups 1.1 Examples In ar
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Groups 11 the triangle ABC into its
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Groups 13 7. Find all symmetries of
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Groups 15 notations (see the soluti
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Groups 17 element, which we will de
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Groups 19 35. Suppose that is the m
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Groups 21 If a group is isomorphic
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Groups 23 tude); (rotation by 180°
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Groups 25 Hence left cosets generat
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Groups 27 as a permutation of the t
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Groups 29 102. Let be the order of
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Groups 31 111. Find all normal subg
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Groups 33 FIGURE 8 127. Prove that
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Groups 35 137. Prove that ker is a
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Groups 37 We now observe what happe
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Groups 39 FIGURE 12 vertices; 4) ro
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Groups 41 Every permutation of degr
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Groups 43 180. Prove that by multip
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Chapter 2 The complex numbers When
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The complex numbers 47 195. Prove t
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The complex numbers 49 where is the
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The complex numbers 51 Consequently
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The complex numbers 53 For complex
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The complex numbers 55 think that n
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The complex numbers 57 214. Let us
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The complex numbers 59 219. Prove t
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The complex numbers 61 functions (s
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The complex numbers 63 of continuit
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The complex numbers 65 erations of
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The complex numbers 67 a) b) c) d)
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The complex numbers 69 REMARK. If a
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The complex numbers 71 2.8 Images o
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The complex numbers 73 where all th
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The complex numbers 75 in such func
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The complex numbers 77 FIGURE 26 no
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The complex numbers 79 C from to th
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The complex numbers 81 DEFINITION.
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The complex numbers 83 FIGURE 32 cu
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The complex numbers 85 to infinity
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The complex numbers 87 property is
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The complex numbers 89 means will b
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The complex numbers 91 For example,
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The complex numbers 93 of the Riema
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The complex numbers 95 in Figure 38
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The complex numbers 97 permutation
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The complex numbers 99 2.13 Monodro
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The complex numbers 101 First we pr
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The complex numbers 103 and prove t
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Chapter 3 Hints, Solutions, and Ans
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Solutions 107 positive integer unde
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Solutions 109 on the left and by on
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Solutions 111 39. a) See Table 8; b
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Solutions 113 of the real numbers i
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Solutions 115 to all and therefore
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Solutions 117 is a subgroup of the
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Solutions 119 hypothesis. The group
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Solutions 121 The given group is th
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Solutions 123 of symmetries of the
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Solutions 125 i.e., there exists an
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Solutions 127 Answer. The normal su
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Solutions 129 In this way the quoti
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Solutions 131 sends every into itse
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Solutions 133 ement of the commutan
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Solutions 135 143. Let be the three
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Solutions 137 150. See 57. Suppose
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Solutions 139 subgroup. Hence if a
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Solutions 141 is commutative. Since
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Solutions 143 180. Since the lower
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Solutions 145 FIGURE 43 190. The pe
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Solutions 147 permutations of type
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Solutions 149 is a field. 195. We h
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Solutions 151 necessarily that and
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Solutions 153 Let C be the field of
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Solutions 155 in as numerators and
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Solutions 157 c) d) e) where 226. W
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Solutions 159 function of real argu
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Solutions 161 i.e., Consider the re
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Solutions 163 Choose as the smalles
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Solutions 165 Since the function is
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Solutions 167 FIGURE 55 FIGURE 56 F
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Solutions 169 FIGURE 60 257. Suppos
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Solutions 171 265. 266. 267. If the
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Solutions 173 following way. If the
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Solutions 175 for all such that and
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Solutions 177 b) In order for to va
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Solutions 179 to the condition coin
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Solutions 181 are continuous functi
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Solutions 183 turn around the other
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Solutions 185 sought is shown in Fi
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Solutions 187 FIGURE 86 FIGURE 87 p
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Solutions 189 sum of multi-valued f
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Solutions 191 316. a) Let and be th
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Solutions 193 by the formal method,
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Solutions 195 The correct scheme is
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Solutions 197 branch of the functio
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Solutions FIGURE 110 FIGURE 111 329
Page 217 and 218: Solutions 201 It follows that and b
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Page 221 and 222: Solutions 205 Since by hypothesis t
Page 223 and 224: Solutions 207 moving along a curve
Page 225 and 226: Drawings of Riemann surfaces 209 Th
Page 227 and 228: FIGURE 119 211
Page 237 and 238: Appendix by A. Khovanskii: Solvabil
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Page 243 and 244: Solvability of Equations 227 where
Page 245 and 246: Solvability of Equations 229 finds
Page 247 and 248: Solvability of Equations 231 quadra
Page 249 and 250: Solvability of Equations 233 More p
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Page 253 and 254: Solvability of Equations 237 Every
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Page 257 and 258: Solvability of Equations 241 FIG. 1
Page 259 and 260: Solvability of Equations 243 are ob
Page 261 and 262: Solvability of Equations 245 corres
Page 265 and 266: Solvability of Equations 249 up to
Page 267: Solvability of Equations 251 fined
Page 271 and 272: Solvability of Equations 255 the no
Page 273 and 274: Solvability of Equations 257 THEORE
Page 277 and 278: Bibliography [1] [2] [3] [4] [5] [6
Page 279 and 280: [23] [24] [25] [26] [27] [28] 263 V
Page 281 and 282: Appendix (V.I. Arnold) The topologi
Page 283 and 284: Index Abel’s theorem, 6, 103 addi
Page 285: pack of sheets, 96 parametric equat
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Abel's theorem in problems and solutions - School of Mathematics

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