Travaux sur les symÃ©tries de Lie des Ã©quations aux ... - DMA - Ens

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220To conclude this extensive introduction which was designed for readerswanting to quickly embrace the contents, we would like to draw theattention on the work [22], whose manual calculations where finalized inmanuscript form already in 2003 24 , and which will soon confirm the abovetheorem by following another route, viz. by calculating explicitly the socalledChern(-Moser) tensor differential forms, which might interest somecontemporary CR geometers better than the (essentially equivalent) originalCartan-Hachtroudi(-Tanaka) approach.§2. SEGRE VARIETIES AND DIFFERENTIAL EQUATIONSReal analytic hypersurfaces in C n+1 . Let us therefore consider an arbitraryreal analytic hypersurface M in C n+1 with n 2, and let us localizeit around one of its points, say p ∈ M. Then there exist complex affinecoordinates:(z, w) = (z 1 , . . .,z n , w) = ( x 1 + √ −1y 1 , . . .,x n + √ −1 y n , u+iv ) = ( x+ √ −1y, u+ √ −1 v )vanishing at p in which T p M = {u = 0}, so that M is represented in aneighborhood of p by a graphed defining equation of the form:u = ϕ(x, y, v) = ϕ ( x 1 , . . .,x n , y 1 , . . .,y n , v ) ,where the real-valued function:∑ϕ = ϕ(x, y, v) =k∈N n , l∈N n , m∈N|k|+|l|+m2ϕ k,l,m x k y l v m ∈ R { x, y, u } ,which possesses entirely arbitrary real coefficients ϕ k,l,m , vanishes at theorigin: ϕ(0) = 0, together with all its first order derivatives: 0 = ∂ x kϕ(0) =∂ y lϕ(0) = ∂ v ϕ(0). By simply rewriting this initial real equation of M as:w+w= ϕ ( z+z, z−z2 2 2 √ , w−w−1 2 −1) √ ,and then by solving the so written equation with respect to w, one obtainsan equation of the shape:w = Θ ( z, z, w ) ∑= Θ k,l,m z k z l w m ∈ C { z, z, w } ,k∈N n , l∈N n , m ∈ N|k|+|l|+m1whose right-hand side converges of course near the origin (0, 0, 0) ∈ C n ×C n × C and whose coefficients Θ k,l,m ∈ C are complex. Since dϕ(0) = 0,one has Θ = −w + order 2 terms.The paradox that any such complex equation provides in fact two realdefining equations for the real hypersurface M which is one-codimensional,24 At the conference Cauchy-Riemann Analysis and Geometry organized by Ingo Lieband Gerd Schmalz at the Max-Planck Institut of Bonn, 22–27 September 2003, the authorgave a talk the title of which was “Explicit Chern-Moser tensors”.
and also in addition the fact that one could as well have chosen to solve theabove equation with respect to w, instead of w, these two apparent “contradictions”are corrected by means of a fundamental, elementary statementthat transfers to Θ (in a natural way) the condition of reality:∑∑ϕ(x, y, u) = ϕ k,l,m x k y l v m = ϕ k,l,m x k y l v m = ϕ(x, y, v)|k|+|l|+m1|k|+|l|+m1enjoyed by the initial definining function ϕ. In the sequel, we shall workexclusively with Θ; the reader is referred to [21] for justifications and motivations.Theorem. ([18], p. 19) The complex analytic function Θ = Θ(z, z, w) withΘ = −w + O(2) together with its complex conjugate:Θ = Θ ( ∑z, z, w) = Θ k,l,m z k z l w m ∈ C { z, z, w }k∈N n , l∈N n , m∈Nsatisfy the two (equivalent by conjugation) functional equations:(7.28)w ≡ Θ ( z, z, Θ(z, z, w) ) ,w ≡ Θ ( z, z, Θ(z, z, w) ) .Conversely, given a local holomorphic function Θ(z, z, w) ∈ C{z, z, w},Θ = −w + O(2) which, in conjunction with its conjugate Θ(z, z, w), satisfiesthis pair of equivalent identities, then the two zero-sets:{ ( )} { ( )}0 = −w + Θ z, z, w and 0 = −w + Θ z, z, wcoincide and define a local one-codimensional real analytic hypersurfaceM passing through the origin in C n+1 .Levi nondegeneracy. Within the hierarchy of nondegeneracy conditionsfor real hypersurfaces initiated by Diederich and Webster ([DW1980], seealso [Me2005a, Me2005b] for generalizations and a unification), Levi nondegeneracyis the most studied. The classical definition may be foundin [Bo1991] and in the survey of Chirka [Ch1991], but the following basicequivalent characterization can also be understood as a definition in thepresent paper. One may show ([Me2005a, Me2005b, 18]) that it is biholomorphicallyinvariant.Lemma. ([18], p. 28) The real analytic hypersurface M ⊂ C n+1 with0 ∈ M represented in coordinates (z 1 , . . .,z n , w) by a complex definingequation of the form w = Θ(z, z, w) is Levi nondegenerate at the origin ifand only if the map:(z1 , . . ., z n , w ) ↦−→(Θ ( 0, z, w ) ,∂Θ∂z 1(0, z, w), . . .,∂Θ∂z n(0, z, w) )221
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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
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The last statements of Corollaries
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[Sha2000] SHAFIKOV, R.: Analytic co
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Here x = (x 1 , . . ., x n ) ∈ K
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Consequently, for the case κ = 2,
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are devoted to provide a general on
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for l = 1, . . .,m such that the lo
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Lemma 8.1. The following conditions
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155namely X ←→ X + X ←→ X .
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In terms of Sussmann’s approach [
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x κ−1 χ κ−1 + O(|x| κ ) + O
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Observe that these expressions are
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where the first term I involves onl
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I 5 = ∑ i 1I 6 = ∑ i 1 ,i 2I 7
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U 2 U κ−1 , we obtain the five f
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Page 173 and 174: linear equations:(7.28) ⎧0 = Rx j
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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 and 192: y means of a fundamental, elementar
Page 193 and 194: 2) When M is Levi nondegenerate at
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: then by replacing the(so obtained)v
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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
Page 243 and 244: Proof. Let l = l ( x i , y j , yβ(
Page 245 and 246: complementary views on the same obj
Page 247 and 248: Classification problem 3.11. Classi
Page 249 and 250: Definition 4.10. L is an infinitesi
Page 251 and 252: Convention 5.2. The letters R will
Page 253 and 254: Assuming that dimSYM(E 1 ) = 8, tak
Page 255 and 256: Restarting from §4.1, let ϕ a Lie
Page 257 and 258: Here, R j l 1 ,kare universal polyn
Page 259 and 260: Lemma 6.34. Let M be a submanifold
Page 261 and 262: where the expressions D β,β1 are
Page 263 and 264: Then the sum L + L is tangent to M
Page 265 and 266: Lemma 7.11. ([CM1974, BER1999, Me20
Page 267 and 268: where(7.33) ∆ := ∑1kn∂ 2∂x
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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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