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192or equivalently, which has a power series expansion of the form:Θ ′( z ′ ,z ′ ,w ′) = − w ′ +∑∑ ∑Θ ′ α,β,0 z′ α z′β + w ′ γΘ ′ α,β,γ z′ α z′β .α1, β1γ1α1, β1Levi nondegenerate hypersurfaces. Leaving aside the real defining equationof M, let us now rename the complex defining equation of M in suchnormalized coordinates simply as before: w = Θ(z, z, w), dropping all theprime signs. Quite concretely, the real analytic hypersurface M is said to beLevi nondegenerate at the origin if the coefficient Θ 1,1,0 of zz, which may bechecked to always be real because of the reality condition (7.28), is nonzero.In fact, it is well known that Levi nondegeneracy is a biholomorphically invariantproperty, see for instance [18], p. 158, but in more conceptual terms,the following general characterization, which may be taken as a definitionhere, holds true. One then readily checks that it is equivalent to Θ 1,1,0 ≠ 0in normalized coordinates.Lemma. ([Me2005a, Me2005b, 19]) The real analytic hypersurface M ⊂C 2 with 0 ∈ M represented in coordinates (z, w) by a complex definingequation of the form w = Θ(z, z, w) is Levi nondegenerate at the origin ifand only if the map:(z,w)↦−→(Θ(0, z, w), Θz (0, z, w) )has nonvanishing 2 × 2 Jacobian determinant at (z, w) = (0, 0).After a possible real dilation of the z-coordinate, we can therefore assumethat Θ 1,1,0 = 1, and then we are provided with the following normalization:(7.28) w = −w + zz + zz O ( |z| + |w| ) ,that will be useful shortly. Another, even more convincing argument forconsigning to oblivion the real defining equation u = ϕ(x, y, v) dates backto Beniamino Segre [23], who observed that to any real analytic M are associatedtwo deeply linked objects.1) The nowadays so-called Segre varieties 15 S q associated to any pointq ∈ C 2 near the origin of coordinates (z q , w q ) that are the complexcurves defined by the equation:S q := { 0 = −w + Θ ( z, z q , w q)},quite appropriately in terms of the fundamental complex defining functionΘ; this equation is holomorphic just because its antiholomorphicterms are set fixed.15 A presentation of the general theory, valuable for generic CR manifolds of arbitrarycodimension d 1 and of arbitrary CR dimension m 1 in C m+d enjoying no specificnondegeneracy condition, may be found in [Me2005a, Me2005b, 18].
2) When M is Levi nondegenerate at the origin, a second-order complexordinary differential equation 16 of the form:w zz (z) = Φ ( z, w(z), w z (z) ) ,193whose solutions are exactly the Segre varieties of M, parametrized bythe two initial conditions w(0) and w z (0) which correspond bijectivelyto the antiholomorphic variables z q and w q .In fact, the recipe for deriving the second-order differential equation associatedto a local Levi-nondegenerate M ⊂ C 2 with 0 ∈ M represented bya normalized 17 equation of the form (7.28) is very simple. Considering thatw = w(z) is given in the equation:w(z) = Θ ( z, z, w )as a function of z with two supplementary (antiholomorphic) parameters zand w that one would like to eliminate, we solve with respect to z and w,just by means of the implicit function theorem 18 , the pair of equations:[w(z) = Θ(z, z, w)= −w + zz + zz O(|z| + |w|)w z (z) = Θ z(z, z, w)= z + z O(|z| + |w|)the second one being obtained by differentiating the first one with respect toz, and this yields a representation:z = ζ ( z, w(z), w z (z) ) and w = ξ ( z, w(z), w z (z) )for certain two uniquely defined local complex analytic functionsζ(z, w, w z ) and ξ(z, w, w z ) of three complex variables. By means ofthese functions, we may then replace z and w in the second derivative:( )w zz (z) = Θ zz z, z, w= Θ zz(z, ζ(z, w(z), wz (z) ) , ξ ( z, w(z), w z (z) ))=: Φ ( z, w(z), w z (z) ) ,and this defines without ambiguity the associated differential equation.More about differential equations will be said in §3 below.16 This idea, usually attributed by contemporary CR geometers to B. Segre, dates in factback (at least) to Chapter 10 of Volume 1 of the 2 100 pages long Theorie der Transformationsgruppenwritten by Sophus Lie and Friedrich Engel between 1884 and 1893, where itis even presented in the uppermost general context.17 In fact, such a normalization was made in advance just in order to make things concreteand clear, but thanks to what the Lemma on p. 192 expresses in a biholomorphicallyinvariant way, everything which follows next holds in an arbitrary system of coordinates.18 Justification: by our preliminary normalization, the 2 × 2 Jacobian determinant∂(Θ, Θ z)∂(z, w)computed at the origin equals˛ 0 −1˛1 0 ˛, hence is nonzero. Without the preliminarynormalization, the condition of the Lemma on p. 192 also applies in any case.
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Joël M E R K E RÉcole Normale Sup
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§1. INTRODUCTIONSeveral physically
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e a collection of m analytic second
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7where the indices j, l 1 vary in {
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This phenomenon could be explained
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This yields the prolongation of the
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X(x, y) and Y (x, y) such that it m
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2.23. Compatibility conditions for
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This lemma is left to the reader; a
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simplification nor any reordering:(
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aleza, y por otra, las organizacion
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computational level (differential-g
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For instance, in the case m = 2, by
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in (3.11). The second equation that
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Similarly, the second equation take
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order derivatives of X and of the Y
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Lemma 3.32. The system yxx j = F j
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Lemma 3.45. The following quadratic
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often be denoted by the sign “·
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1Now, taking account of the factor
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conditions, totally equivalent to t
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43remaining terms afterwards:(3.71)
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Here, the sign ≡ precisely means:
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Next, replacing plainly (3.64) in (
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49the order of §3.73. We get:(3.89
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51Multiplying by −2 and reorganiz
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⎧Θ l 1y l 2 = −Ll 1l 1 ,l 1 ,y
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+ 1 ∑4 δj l 1Hl k 2L k k,kk− 2
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+ 1 ∑2 δj l 1Hl k 3M l2 ,kk+ 1 4
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In the hardest techical part of thi
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− 1 3+ 1 3− 1 4+ 1 4∑Hl k 1H
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− 1 3∑kL l 1l1 ,k,x Hk l 1+ 1 3
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Here, the sign ≡ means “modulo
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Thirdly, put j := l 2 in (3.108) wi
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69correct. We get:(4.29)0 =?== −
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Next, apply the operator ∑ k Ll 2
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73Writing term by term the substrac
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of the terms of the subgoal (4.29):
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77+ 1 2∑kH k l 1 ,y l 2 Hl 2k15
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79= − X xx Yx j + Y jm∑+++++l 1
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Our first task is to compute the de
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Replacing this expression of A k in
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85have the continuation(5.17) −y
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87and where thirdly (we are nearly
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89obtain:(5.23)III :=m∑m∑l 1 =1
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[GTW1989] GRISSOM, C.; THOMPSON, G.
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93Nonalgebraizable real analytic tu
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and of CR dimension m = n − d ≥
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coordinates z ′ = 2i ln(z/z p ),
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{x ∈ K n : |x| < ρ} for some ρ
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(2) A K-algebraic inversion mapping
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(1) The complex Lie algebra Hol(M,
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Proposition 3.1. Let t ↦→ G i (
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The first relation gives nothing, s
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with |β∗ k | ≥ 1 and integers
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Clearly, the left hand side is an a
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field X 1 ′ := h ∗ (X 1 ). Taki
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which yields after differentiating
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117Im w∆ n(ρ 4 )∆ n(ρ 1 )∆
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[∂ i,ei H ei (t ′ )] t ′ =He
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p = (w p , z p , ζ p , ξ p ) ∈
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Finite nondegeneracy is interesting
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such that |ω j i ∗ ,β∗ i(t,
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for k even, we have Γ k (z (k) ) =
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that for every local holomorphic se
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h(t) = ˜H(t, J l 0µ 0¯h(0)) with
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infinite families of pairwise non b
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functions w(z)). The coefficients R
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Then the Lie criterion states that
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y k ′ = λk k y k and w ′ = µ
Page 141 and 142: The last statements of Corollaries
Page 143 and 144: [Sha2000] SHAFIKOV, R.: Analytic co
Page 145 and 146: Here x = (x 1 , . . ., x n ) ∈ K
Page 147 and 148: Consequently, for the case κ = 2,
Page 149 and 150: are devoted to provide a general on
Page 151 and 152: for l = 1, . . .,m such that the lo
Page 153 and 154: Lemma 8.1. The following conditions
Page 155 and 156: 155namely X ←→ X + X ←→ X .
Page 157 and 158: In terms of Sussmann’s approach [
Page 159 and 160: x κ−1 χ κ−1 + O(|x| κ ) + O
Page 161 and 162: Observe that these expressions are
Page 163 and 164: where the first term I involves onl
Page 165 and 166: I 5 = ∑ i 1I 6 = ∑ i 1 ,i 2I 7
Page 167 and 168: U 2 U κ−1 , we obtain the five f
Page 169 and 170: and U 2 U κ−1 , we obtain the fi
Page 171 and 172: following equations for j = 1, . .
Page 173 and 174: linear equations:(7.28) ⎧0 = Rx j
Page 175 and 176: 175R j containing at least one part
Page 177 and 178: 177[3] :∑σ∈S κ−1κδ l,....
Page 179 and 180: the values of the partial derivativ
Page 181 and 182: also appears some derivatives Q l
Page 183 and 184: [5] F. ENGEL; LIE, S.: Theorie der
Page 185 and 186: 185Nonrigid sphericalreal analytic
Page 187 and 188: origin:{ ( ∣ ∣ )AJ 4 1∣∣∣
Page 189 and 190: I 2 characterizes equivalence to w
Page 191: y means of a fundamental, elementar
Page 195 and 196: 195holding in C { z ′ , z ′ , w
Page 197 and 198: Fortunately for our present purpose
Page 199 and 200: We notice passim that S ≡ Q x (no
Page 201 and 202: for a certain local K-analytic new
Page 203 and 204: Conversely, if y xx (x) = F ( x, y(
Page 205 and 206: 205But we may also express the dual
Page 207 and 208: Then thanks to a straightforward ap
Page 209 and 210: and we then expand carefully the re
Page 211 and 212: explicit computation, so let us rew
Page 213 and 214: 213+T ah3Q 2 a Q b∆(aa|b)∆(b|bb
Page 215 and 216: 215Vanishing Hachtroudi curvaturean
Page 217 and 218: to fix the ideas, it will be assume
Page 219 and 220: then by replacing the(so obtained)v
Page 221 and 222: and also in addition the fact that
Page 223 and 224: and this defines without ambiguity
Page 225 and 226: the leaves of this local foliation
Page 227 and 228: Then by differentiating with respec
Page 229 and 230: ∂Here, the coefficients of the2 T
Page 231 and 232: 231for (7.28):( )∂□ ·∂a □
Page 233 and 234: 233Lie symmetriesof partial differe
Page 235 and 236: Differentiating the first equation
Page 237 and 238: Example 1.21. (Continued) With n =
Page 239 and 240: defined by the graphed equations:((
Page 241 and 242: for some two local analytic maps Π
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Proof. Let l = l ( x i , y j , yβ(
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complementary views on the same obj
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Classification problem 3.11. Classi
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Definition 4.10. L is an infinitesi
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Convention 5.2. The letters R will
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Assuming that dimSYM(E 1 ) = 8, tak
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Restarting from §4.1, let ϕ a Lie
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Here, R j l 1 ,kare universal polyn
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Lemma 6.34. Let M be a submanifold
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where the expressions D β,β1 are
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Then the sum L + L is tangent to M
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Lemma 7.11. ([CM1974, BER1999, Me20
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where(7.33) ∆ := ∑1kn∂ 2∂x
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with r, s being unspecified functio
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and moreover, all other, higher ord
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To conclude, we replace X xx so obt
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Definition 8.43. A real analytic hy
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Lemma 8.59. The function b depends
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where the letter r denotes an unspe
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(8.80) ⎧Y 1,1,1 = Y x 1 x 1 x 1 +
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(X 1 , X 2 , Y , Xy 1,X 2x, Y 2 x 1
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espect to x 1 and (8.96) 2 with res
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For the two unknowns Xyy 1 and X 2x
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Redeveloping the determinant, the v
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291are obtained by equating to zero
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Let (z 0 , c 0 ) = (x 0 , y 0 , a 0
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we deduce that Γ 5 is submersive (
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11.4. Regularity and jet parametriz
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Cramer’s rule, we get(11.13) ⎧
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in K[a, z] n+m and in K[x, c] p+m .
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of the cancellation properties (11.
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305II: Explicit prolongations of in
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Define then the transformed jet ϕ
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Let λ ∈ N be an arbitrary intege
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311[Ol1986], [BK1989]):(Y(1.31)⎧
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+ [Y y 2 − 2 X xy ] y 1 y 2 + [
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for some nonnegative integers A, B,
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Once the correct theorem is formula
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By the classical formula for the de
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The induction formulas are⎧(3.4)
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We have to compute:( ) ⎛ ⎞∑
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Next, we gather the underlined term
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+ ∑k 1 ,k 2 ,k 3 ,k 4[−δ k 1,k
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Correspondingly, we identify the se
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must be equal to the number of indi
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Y i1 ,i 2 ,i 3 ,i 4written in one o
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335conclusion, we have shown that (
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337Theorem 3.73. For every κ 1 an
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Theorem 3.79. For every integer κ
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+++m∑l 1 ,l 2 =1m∑l 1 ,l 2 ,l 3
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343form:(4.12)κ+1Yκ j = Y jx κ +
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eing related to the number 2 in the
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Appying the chain rule, we may comp
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Secondly:(5.3)Y j i 1 ,i 2= Y jx i
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As explained before the statement o
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g j i 1 ,...,i λand similarly for
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with respect to the variables x i 1
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Assuming F = F(x, y x ) to be indep
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359(III’) ⎧0 = ∑ (δ k 2)j 3H
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Open problem 1.17. For n = 2 establ
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⎧∣ ⎨X 1 xX 1+ y x 1 y x 1 ·1
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eplacing the third column by the se
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⎧ ∣ ∣⎫ ⎨ ∣∣∣∣∣
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e written under the specific form:(
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Indeed, the collection of equations
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3.12. Principal unknowns. As there
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375Sixthly:(3.22) ⎧δ k 1j 1Θ n+
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are a consequence of (I’), (I”)
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379IV: BibliographyREFERENCES[Ar198
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[G1989] GARDNER, R.B.: The method o
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[Me2005a] MERKER, J.: On the local
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