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

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348we observe that the following induction relations hold:(4.23)h 2 = F 2 (h 1 ) ,h 3 = F 3 (h 2 ) ,. . . . . . . . . . . . . . .h λ = F λ (h λ−1 ) .To obtain the explicit version of the Faà di Bruno in the case of one variable xand several variables (y 1 , . . ., y m ), it suffices to extract from the expressionof Y j κ provided by Theorem 4.18 only the terms corresponding to µ 1λ 1 +· · · + µ d λ d = κ, dropping all the X terms. After some simplifications andafter a translation by means of an elementary dictionary, we may formulatea statement.Theorem 4.24. For every integer κ 1, the κ-th partial derivative of thecomposite function h = h(x) = f (g 1 (x), . . .,g m (x)) with respect to x maybe expressed as an explicit polynomial depending on the partial derivativesof f, on the derivatives of g and having integer coefficients:(4.25)d κ hκ∑dx = κd=1∑1λ 1
Secondly:(5.3)Y j i 1 ,i 2= Y jx i 1x i 2 + m ∑++++m∑l 1 ,l 2 =1m∑l 1 ,l 2 ,l 3 =1m∑l 1 =1m∑l 1 ,l 2 =1l 1 =1n∑k 1 ,k 2 =1n∑k 1 ,k 2 =1n∑k 1 =1[]δ k 1i 1Y jx i 2y + l 1 δk 1i 2Y jx i 1y − l 1 δj l 1X k 1y l x i 1x i 12 k1+349[]δ k 1,k 2i 1 , i 2Y jy l 1y − l 2 δj l 2δ k 1i 1X k 2x i 2y − l 1 δj l 2δ k 1i 2X k 2y l 1x i 1y l 1 k1y l 2k2+n∑k 1 ,k 2 ,k 3 =1[]−δ j l 3δ k 1,k 2i 1 , i 2X k 3y l y l 1y l 12 k1y l 2k2y l 3k3+[]δ k 1,k 2i 1 , i 2Y jy − l 1 δj l 1δ k 1i 1X k 2x − i 2 δj l 1δ k 1i 2X k 2y l x i 11 k1 ,k 2+n∑k 1 ,k 2 ,k 3 =1[]−δ j l 1δ k 2,k 3i 1 , i 2X k 1y − l 2 δj l 2δ k 3,k 1i 1 , i 2X k 2y − l 1 δj l 2δ k 1,k 2i 1 , i 2X k 3y l y l 11 k1y l 2k2y l 3k3.Thirdly:Y j i 1 ,i 2 ,i 3= Y jx i 1x i 2x i 3 + m ∑++m∑l 1 ,l 2 =1m∑l 1 ,l 2 ,l 3 =1l 1 =1n∑k 1 ,k 2 =1n∑k 1 =1[]δ k 1i 1Y jx i 2x i 3y + l 1 δk 1i 2Y jx i 1x i 3y + l 1 δk 1i 3Y jx i 1x i 2y − l 1 δj l 1X k 1y l x i 1x i 2x i 13 k1+[δ k 1,k 2i 1 , i 2Y jx i 3y l 1y l 2 + δk 1,k 2i 3 , i 1Y jx i 2y l 1y l 2 + δk 1,k 2i 2 , i 3Y jx i 1y l 1y l 2 −n∑k 1 ,k 2 ,k 3 =1]−δ j l 2δ k 1i 1X k 2x i 2x i 3y − l 1 δj l 2δ k 1i 2X k 2x i 1x i 3y − l 1 δj l 2δ k 1i 3X k 2y l x i 1x i 2y l 11 k1y l 2[δ k 1,k 2 ,k 3i 1 , i 2 , i 3Y jy l 1y l 2y − l 3 δj l 3δ k 1,k 2i 1 , i 2X k 3x i 3y l 1y − l 2]−δ j l 3δ k 1,k 2i 1 , i 3X k 3x i 2y l 1y − l 2 δj l 3δ k 1,k 2i 2 , i 3X k 3y l x i 1y l 1y l 12 k1y l 2k2y l 3k3+k2+++m∑l 1 ,l 2 ,l 3 ,l 4 =1m∑l 1 =1n∑k 1 ,k 2 =1n∑k 1 ,k 2 ,k 3 ,k 4 =1[]−δ j l 4δ k 1,k 2 ,k 3i 1 , i 2 , i 3X k 4y l y l 1y l 2y l 13 k1y l 2k2y l 3k3y l 4k4+[δ k 1,k 2i 1 , i 2Y jx i 3y l 1 + δk 1,k 2i 3 , i 1Y jx i 2y l 1 + δk 1,k 2i 2 , i 3Y jx i 1y l 1 −]−δ j l 1δ k 1i 1X k 2x i 2x − i 3 δj l 1δ k 1i 2X k 2x i 1x − i 3 δj l 1δ k 1i 3X k 2y l x i 1x i 12 k1 ,k 2+
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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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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 (
Page 297 and 298: 11.4. Regularity and jet parametriz
Page 299 and 300: Cramer’s rule, we get(11.13) ⎧
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Page 303 and 304: of the cancellation properties (11.
Page 305 and 306: 305II: Explicit prolongations of in
Page 307 and 308: Define then the transformed jet ϕ
Page 309 and 310: Let λ ∈ N be an arbitrary intege
Page 311 and 312: 311[Ol1986], [BK1989]):(Y(1.31)⎧
Page 313 and 314: + [Y y 2 − 2 X xy ] y 1 y 2 + [
Page 315 and 316: for some nonnegative integers A, B,
Page 317 and 318: Once the correct theorem is formula
Page 319 and 320: By the classical formula for the de
Page 321 and 322: The induction formulas are⎧(3.4)
Page 323 and 324: We have to compute:( ) ⎛ ⎞∑
Page 325 and 326: Next, we gather the underlined term
Page 327 and 328: + ∑k 1 ,k 2 ,k 3 ,k 4[−δ k 1,k
Page 329 and 330: Correspondingly, we identify the se
Page 331 and 332: must be equal to the number of indi
Page 333 and 334: Y i1 ,i 2 ,i 3 ,i 4written in one o
Page 335 and 336: 335conclusion, we have shown that (
Page 337 and 338: 337Theorem 3.73. For every κ 1 an
Page 339 and 340: Theorem 3.79. For every integer κ
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Page 343 and 344: 343form:(4.12)κ+1Yκ j = Y jx κ +
Page 345 and 346: eing related to the number 2 in the
Page 347: Appying the chain rule, we may comp
Page 351 and 352: As explained before the statement o
Page 353 and 354: g j i 1 ,...,i λand similarly for
Page 355 and 356: with respect to the variables x i 1
Page 357 and 358: Assuming F = F(x, y x ) to be indep
Page 359 and 360: 359(III’) ⎧0 = ∑ (δ k 2)j 3H
Page 361 and 362: Open problem 1.17. For n = 2 establ
Page 363 and 364: ⎧∣ ⎨X 1 xX 1+ y x 1 y x 1 ·1
Page 365 and 366: eplacing the third column by the se
Page 367 and 368: ⎧ ∣ ∣⎫ ⎨ ∣∣∣∣∣
Page 369 and 370: e written under the specific form:(
Page 371 and 372: Indeed, the collection of equations
Page 373 and 374: 3.12. Principal unknowns. As there
Page 375 and 376: 375Sixthly:(3.22) ⎧δ k 1j 1Θ n+
Page 377 and 378: are a consequence of (I’), (I”)
Page 379 and 380: 379IV: BibliographyREFERENCES[Ar198
Page 381 and 382: [G1989] GARDNER, R.B.: The method o
Page 383 and 384: [Me2005a] MERKER, J.: On the local
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