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1623.3. Computation of R κ in the general case. Following the exact samescheme as in the case n = 1 we give the general partial formula forR κ . We start with the first three families of coefficients R j k 1, R j k 1 ,k 2andR j k 1 ,k 2 ,k 3. Let δp q be the Kronecker symbol, equal to 1 if p = q and to 0 ifp ≠ q. More generally, the generalized Kronecker symbols are defined byδ q 1,...,q kp 1 ,...,p k:= δ q 1p 1δ q 2p 2 · · ·δ q kp k.By convention, the indices j, i 1 , i 2 , . . . , i λ run in the set {1, . . ., m}, theindices k, k 1 , k 2 , . . . , k λ and l, l 1 , l 2 , . . . , l λ running in {1, . . ., n}. Hence wewill write ∑ m ∑ mi 1 =1 i 2 =1 · · ·∑mi λ =1 as ∑ i 1 ,...,i λand ∑ n ∑ nl 1 =1 l 2 =1 · · ·∑nl λ =1as ∑ l 1 ,...,l λ. The letters i 1 , i 2 , . . .,i λ and l 1 , l 2 , . . .,l λ will always be used forthe summations in the development of R j k 1 ,k 2 ,...,k λ. We will always use theindices j and k 1 , k 2 , . . ., k λ to write the coefficient R j k 1 ,k 2 ,...,k λ.We have:(7.28)⎧R ⎪⎨j k 1= Rx j k1+ ∑ i 1⎪⎩∑ []δ l 1k1R j − u i 1 δj i 1Q l 1xk1U i 1l 1+l 1+ ∑ ∑i 1 ,i 2l 1 ,l 2[−δji 2δ l 1k1Q l 2u i 1]Ui 1l 1U i 2l 2.For R j k 1 ,k 2we have:(7.28) ⎧R j k 1 ,k 2= Rx j k1 x k2+ ∑ i 1+ ∑ ∑ [(δ l 1,l 2k 1 ,k 2R j u i 1u − i 2 δj i 2δ l 1k1Q l 2+ x k2 u i 1 δl 1k2Q l 2x k1 u i 1i 1 ,i 2 l 1 ,l 2⎪⎨+ ∑⎪⎩i 1 ,i 2 ,i 3∑ [−δ j i 3δ l 1,l 2k 1 ,k 2Q l 3u i 1u i 2l 1 ,l 2 ,l 3∑l 1[δ l 1k2R j x k1 u i 1 + δl 1k1R j x k2 u i 1 − δj i 1Q l 1xk1 x k2]U i 1l 1+]U i 1l 1U i 2l 2U i 3l 3++ ∑ ∑ []δ l 1,l 2k 1 ,k 2R j − u i 1 δj i 1δ l 1k2Q l 2xk1− δ j i 1δ l 1k1Q l 2xk2U i 1l 1 ,l 2+i 1 l 1 ,l 2+ ∑ i 1 ,i 2∑l 1 ,l 2 ,l 3[−δ j i 2δ l 1,l 2k 1 ,k 2Q l 3u i 1 − δj i 2δ l 3,l 1k 1 ,k 2Q l 2u i 1 − δj i 1δ l 2,l 3k 1 ,k 2Q l 1u i 2)]U i 1l 1U i 2l 2+]U i 1l 1U i 2l 2 ,l 3.Since we also treat systems of order κ ≥ 3, it is necessary to computeR j k 1 ,k 2 ,k 3. We write this as follows:(7.28) R j k 1 ,k 2 ,k 3= I + II + III,
where the first term I involves only polynomials in U i 1(7.28) ⎧I = Rx j k1 x k2 x k3+ ∑ ∑ [δ l 1k1R j + x k2 x k3 u i 1 δl 1k2R j + x k1 x k3 u i 1 δl 1k3R j − x k1 x k2 u i 1i 1 l 1]−δ j i 1Q l 1xk1 x k2 x k3U i 1l 1+ ∑ ∑ [δ l 1,l 2k 1 ,k 2R j x k3 u i 1u + i 2 δl 1,l 2k 3 ,k 1R j x k2 u i 1u + i 2i 1 ,i 2 l 1 ,l 2⎪⎨⎪⎩+δ l 1,l 2k 2 ,k 3R j x k1 u i 1u − i 2 δj i 2δ l 1k1Q l 2]−δ j i 2δ l 1k3Q l 2x k1 x k2U i u i 11 l 1U i 2l 2+ ∑−δ j i 3δ l 1,l 2k 1 ,k 2Q l 3−δ j i 3δ l 1,l 2k 1 ,k 3Q l 3+ ∑i 1 ,i 2 ,i 3 ,i 4x k3 u i 1u i 2 − δj i 3δ l 1,l 2x k2 u i 1u i 2x k2 x k3 u i 1 − δj i 2δ l 1i 1 ,i 2 ,i 3k 2 ,k 3Q l 3x k1 u i 1u − i 2]U i 1l 1U i 2l 2U i 3l 3+∑ [−δ j i 4δ l 1,l 2 ,l 3k 1 ,k 2 ,k 3Q l 4u i 1u i 2u i 3l 1 ,l 2 ,l 3 ,l 4l 1:k2Q l 2− x k1 x k3 u i 1∑ [δ l 1,l 2 ,l 3k 1 ,k 2 ,k 3R j u i 1u i 2u − i 3l 1 ,l 2 ,l 3]U i 1l 1U i 2l 2U i 3l 3U i 4l 4,163the second term II involves at least once the monomial U i 1l 1 ,l 2:(7.28)⎧II = ∑ i 1⎪⎨⎪⎩∑ [δ l 1,l 2k 1 ,k 2R j + x k3 u i 1 δl 1,l 2k 3 ,k 1R j + x k2 u i 1 δl 1,l 2k 2 ,k 3R j − x k1 u i 1l 1 ,l 2)]−δ j i 1(δ l 1k1Q l 2xk2 x k3+ δ l 1k2Q l 2xk1 x k3+ δ l 1k3Q l 2xk1 x k2U i 1l 1 ,l 2++ ∑ ∑ [δ l 1,l 2 ,l 3k 1 ,k 2 ,k 3R j u i 1u + i 2 δl 3,l 1 ,l 2k 1 ,k 2 ,k 3R j u i 1u + i 2 δl 2,l 3 ,l 1k 1 ,k 2 ,k 3R j u i 1u − i 2i 1 ,i 2 l 1 ,l 2 ,l 3−δ j i 1(δ l 2,l 3k 1 ,k 2Q l 1x k3 u i 2 + δl 2,l 3k 3 ,k 1Q l 1x k2 u i 2 + δl 2,l 3k 2 ,k 3Q l 1x k1 u i 2−δ j i 2(δ l 1,l 2k 1 ,k 2Q l 3x k3 u i 1 + δl 1,l 2k 3 ,k 1Q l 3x k2 u i 1 + δl 1,l 2k 2 ,k 3Q l 3+δ l 3,l 1k 1 ,k 2Q l 2+ ∑i 1 ,i 2 ,i 3+ x k3 u i 1 δl 3,l 1k 3 ,k 1Q l 2+ x k2 u i 1 δl 3,l 1k 2 ,k 3Q l 2x k1 u i 1x k1 u i 1 +)−)]U i 1l 1U i 2l 2 ,l 3+∑ [−δ j i 3(δ l 1,l 2 ,l 3k 1 ,k 2 ,k 3Q l 4u i 1u + i 2 δl 1,l 4 ,l 2k 1 ,k 2 ,k 3Q l 3u i 1u + i 2l 1 ,l 2 ,l 3 ,l 4) (− δ j i 1δ l 3,l 1 ,l 2k 1 ,k 2 ,k 3Q l 4u i 1u i 2+δ l 2,l 3 ,l 4k 1 ,k 2 ,k 3Q l 1u i 1u i 2δ l 3,l 2 ,l 4k 1 ,k 2 ,k 3Q l 1u i 2u i 3 + δl 4,l 3 ,l 2k 1 ,k 2 ,k 3Q l 1)]U i 1l 1U i 2l 2U i 3l 3 ,l 4+ ∑ i 1 ,i 2+δ l 3,l 1 ,l 2k 1 ,k 2 ,k 3Q l 4u i 1 + δl 2,l 3 ,l 1k 1 ,k 2 ,k 3Q l 4u i 1)]U i 1l 1 ,l 2U i 2l 3 ,l 4u i 2u i 3 +∑ [−δ j i 2(δ l 1,l 2 ,l 3k 1 ,k 2 ,k 3Q l 4+ u i 1l 1 ,l 2 ,l 3 ,l 4
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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
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
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: Observe that these expressions are
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 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
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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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