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

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100identify X and X + X and say by some abuse of language that X itself isan infinitesimal CR automorphism.In the algebraic category, the main drawback of infinitesimal CR automorphismis that they do not have algebraic flow. For instance, the complexdilatation vector field X = iz ∂ z has transcendent flow, even if it is an infinitesimalCR automorphism of every algebraic hypersurface in C 2 whoseequation is of the form v = ϕ(z¯z), even if the coefficient of X is algebraic.Thus instead of infinitesimal CR automorphisms which generate oneparametergroups of biholomorphic transformations of M, we shall studyalgebraically dependent one-parameter families of biholomorphic transformations(not necessarily making a one parameter group). To begin with, weneed to introduce some precise definitions about local algebraic Lie transformationgroups.2.3. Local Lie group actions in the K-algebraic category. Often in real orin complex analytic geometry, the interest cannot be focalized on global Lietransformation groups, but only on local transformations which are close tothe identity. For instance, the transformation group of a small piece of a realanalytic CR manifold in C n which is not contained in a global, large or compactCR manifold is almost never a true, global transformation group. Consequentlythe usual axioms of Lie transformation groups must be localized.Philosophically speaking, the local point of view is often the most adequateand the richest one, because a given analytico-geometric object often possessesmuch more local invariant than global invariants, if any. Historicallyspeaking, the local Lie transformation groups were first studied, before theintroduction of the now classical notion of global Lie group. Especially, inhis first masterpiece work [Lie1880] on the subject, Sophus Lie essentiallydealt with local “Lie” groups: he classified all continuous local transformationgroups acting on an open subset of C 2 . This general classificationprovided afterwards in the years 1880–1890 many applications to the localstudy of differential equations: local normal forms, local solvability, etc.In this paragraph we define precisely local actions of local Lie groups andwe focus especially on the K-algebraic category.Let c ∈ N ∗ , let g = (g 1 , . . .,g c ) ∈ K c and let two positive numberssatisfy 0 < δ 2 < δ 1 . We formulate the desired definition by means of thetwo precise polydiscs ∆ c (δ 2 ) ⊂ ∆ c (δ 1 ) ⊂ K c . A local K-algebraic Liegroup of dimension c consists of the following data:(1) A K-algebraic multiplication mapping µ : ∆ c (δ 2 ) × ∆ c (δ 2 ) → ∆ c (δ 1 )which is locally associative (µ(g, µ(g ′ , g ′′ )) = µ(µ(g, g ′ ), g ′′ )), wheneverµ(g ′ , g ′′ ) ∈ ∆ c (δ 2 ), µ(g, g ′ ) ∈ ∆ c (δ 2 ) and which satisfiesµ(0, g) = µ(g, 0) = g, where the origin 0 ∈ K c corresponds to theidentity element in the group structure.
(2) A K-algebraic inversion mapping ι : ∆ c (δ 2 ) → ∆ c (δ 1 ) satisfyingµ(g, ι(g)) = µ(ι(g), g) = 0 and ι(0) = 0 whenever ι(g) ∈ ∆ c (δ 2 ).Here, the integer c ∈ N ∗ is the dimension of G. We shall say that compositionand inversion are defined locally in a neighborhood of the identityelement. In the K-analytic category, the corresponding definition is similar.Now, we can define the notion of local K-algebraic Lie group action. Letn ∈ N ∗ , let x = (x 1 , . . ., x n ) ∈ K n and let two positive numbers satisfy 0
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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 (
Page 49 and 50: 49the order of §3.73. We get:(3.89
Page 51 and 52: 51Multiplying by −2 and reorganiz
Page 53 and 54: ⎧Θ l 1y l 2 = −Ll 1l 1 ,l 1 ,y
Page 55 and 56: + 1 ∑4 δj l 1Hl k 2L k k,kk− 2
Page 57 and 58: + 1 ∑2 δj l 1Hl k 3M l2 ,kk+ 1 4
Page 59 and 60: In the hardest techical part of thi
Page 61 and 62: − 1 3+ 1 3− 1 4+ 1 4∑Hl k 1H
Page 63 and 64: − 1 3∑kL l 1l1 ,k,x Hk l 1+ 1 3
Page 65 and 66: Here, the sign ≡ means “modulo
Page 67 and 68: Thirdly, put j := l 2 in (3.108) wi
Page 69 and 70: 69correct. We get:(4.29)0 =?== −
Page 71 and 72: Next, apply the operator ∑ k Ll 2
Page 73 and 74: 73Writing term by term the substrac
Page 75 and 76: of the terms of the subgoal (4.29):
Page 77 and 78: 77+ 1 2∑kH k l 1 ,y l 2 Hl 2k15
Page 79 and 80: 79= − X xx Yx j + Y jm∑+++++l 1
Page 81 and 82: Our first task is to compute the de
Page 83 and 84: Replacing this expression of A k in
Page 85 and 86: 85have the continuation(5.17) −y
Page 87 and 88: 87and where thirdly (we are nearly
Page 89 and 90: 89obtain:(5.23)III :=m∑m∑l 1 =1
Page 91 and 92: [GTW1989] GRISSOM, C.; THOMPSON, G.
Page 93 and 94: 93Nonalgebraizable real analytic tu
Page 95 and 96: and of CR dimension m = n − d ≥
Page 97 and 98: coordinates z ′ = 2i ln(z/z p ),
Page 99: {x ∈ K n : |x| < ρ} for some ρ
Page 103 and 104: (1) The complex Lie algebra Hol(M,
Page 105 and 106: Proposition 3.1. Let t ↦→ G i (
Page 107 and 108: The first relation gives nothing, s
Page 109 and 110: with |β∗ k | ≥ 1 and integers
Page 111 and 112: Clearly, the left hand side is an a
Page 113 and 114: field X 1 ′ := h ∗ (X 1 ). Taki
Page 115 and 116: which yields after differentiating
Page 117 and 118: 117Im w∆ n(ρ 4 )∆ n(ρ 1 )∆
Page 119 and 120: [∂ i,ei H ei (t ′ )] t ′ =He
Page 121 and 122: p = (w p , z p , ζ p , ξ p ) ∈
Page 123 and 124: Finite nondegeneracy is interesting
Page 125 and 126: such that |ω j i ∗ ,β∗ i(t,
Page 127 and 128: for k even, we have Γ k (z (k) ) =
Page 129 and 130: that for every local holomorphic se
Page 131 and 132: h(t) = ˜H(t, J l 0µ 0¯h(0)) with
Page 133 and 134: infinite families of pairwise non b
Page 135 and 136: functions w(z)). The coefficients R
Page 137 and 138: Then the Lie criterion states that
Page 139 and 140: 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
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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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and U 2 U κ−1 , we obtain the fi
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following equations for j = 1, . .
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linear equations:(7.28) ⎧0 = Rx j
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175R j containing at least one part
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177[3] :∑σ∈S κ−1κδ l,....
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the values of the partial derivativ
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also appears some derivatives Q l
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[5] F. ENGEL; LIE, S.: Theorie der
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185Nonrigid sphericalreal analytic
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origin:{ ( ∣ ∣ )AJ 4 1∣∣∣
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I 2 characterizes equivalence to w
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y means of a fundamental, elementar
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2) When M is Levi nondegenerate at
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195holding in C { z ′ , z ′ , w
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Fortunately for our present purpose
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We notice passim that S ≡ Q x (no
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for a certain local K-analytic new
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Conversely, if y xx (x) = F ( x, y(
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205But we may also express the dual
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Then thanks to a straightforward ap
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and we then expand carefully the re
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explicit computation, so let us rew
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213+T ah3Q 2 a Q b∆(aa|b)∆(b|bb
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215Vanishing Hachtroudi curvaturean
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to fix the ideas, it will be assume
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then by replacing the(so obtained)v
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and also in addition the fact that
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and this defines without ambiguity
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the leaves of this local foliation
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Then by differentiating with respec
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∂Here, the coefficients of the2 T
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231for (7.28):( )∂□ ·∂a □
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233Lie symmetriesof partial differe
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Differentiating the first equation
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Example 1.21. (Continued) With n =
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defined by the graphed equations:((
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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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