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

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366⎧ ∣ ∣⎫ ⎨ ∣∣∣∣∣ X 1+ y x 1 y x 1 ·⎩ − x 2 yX 1 xX 1 2 y ∣∣∣∣∣ ⎬X 2 x 2 yX 2 xX 2 2 y⎭ +Y x 2 y Y x 2 Y y⎧∣ ∣⎨X 1 xX 1+ y x 1 y x 2 ·1 xX 1 2 yy ∣∣∣∣∣ X 1⎩X 2 xX 2∣1 xX 2 x 2 yy −1 yX 1 xX 2 y1 ∣∣∣∣∣ X 2 xY x 1 Y x 2 Y yy∣1 yX 2 xX 2 2 y −Y x 1 y Y x 2 Y y ∣⎫ ⎧ ∣ ∣⎫ X 1 xX 1−1 x 2 yXy1 ∣∣∣∣∣ ⎬ ⎨ ∣∣∣∣∣ X 1 X 2 xX 2∣1 x 2 yXy2 ⎭ + y x 2 y x 2 ⎩ − xX 1 1 x 1 yXy1 ∣∣∣∣∣ ⎬X 2 xX 2 1 x 1 yXy2 ⎭ +Y x 1 Y x 2 y Y y Y x 1 Y x 1 y Y y⎧ ∣ ∣⎫ ⎨ ∣∣∣∣∣ X 1+ y x 1 y x 1 y x 2 ·⎩ − yy X 1 xX 1 2 y ∣∣∣∣∣ ⎬Xyy 2 X 2 xX 2 2 y⎭ +Y yy Y x 2 Y y⎧ ∣ ∣⎫ ⎨ ∣∣∣∣∣ X 1+ y x 1 y x 2 y x 2 ·⎩ − xX 1 1 yy Xy1 ∣∣∣∣∣ ⎬X 2 xX 2 1 yy Xy2 ⎭ .Y x 1 Y yy Y yUsing the third equation (2.15), we also deduce, without any further computation,that the third equation which would be obtained by developing (2.9)in length, is:(2.23) ∣ ∣ ∣ ∣∣∣∣∣ X 10 = y x 2 x 2 · xX 1 1 xX 1 2 y ∣∣∣∣∣ X 1X 2 xX 2 1 xX 2 xX 12 y +1 xX 1 ∣∣∣∣∣ 2 x 2 x 2X 2 xX 2Y x 1 Y x 2 Y y∣1 xX 2 2 x 2 x+ 2Y x 1 Y x 2 Y x 2 x⎧ ∣ ⎫2⎨ ∣∣∣∣∣ X 1+ y x 1 ·⎩ − x 2 xX 1 2 xX 1 2 y⎬X 2 x 1 xX 2 2 xX 2 2 yY x 2 x 2 Y x 2 Y ∣⎭ +y⎧ ∣ ∣ ⎫⎨ ∣∣∣∣∣ X 1+ y x 2 ·⎩ 2 xX 1 1 xX 1 2 x 2 y ∣∣∣∣∣ X 1X 2 xX 2 1 xX 2 2 x 2 yY x 1 Y x 2 Y x∣ − xX 1 1 x 2 xX 1 2 y⎬X 2 xX 2 1 x 2 xX 2 2 y2 y Y x 1 Y x 2 x 2 Y ∣⎭ +y
⎧ ∣ ∣⎫ ⎨ ∣∣∣∣∣ X 1+ y x 1 y x 2 ·⎩ −2 x 2 yX 1 xX 1 2 y ∣∣∣∣∣ ⎬X 2 x 2 yX 2 xX 2 2 y⎭ +Y x 2 y Y x 2 Y y⎧∣ ⎨X 1 xX 1+ y x 2 y x 2 ·1 xX 2 yy1 ∣∣∣∣∣ ⎩X 2 xX 2∣1 xX 2 2 yy − 2Y x 1 Y x 2 Y yy∣⎧ ∣ ∣⎫ ⎨ ∣∣∣∣∣ X 1+ y x 1 y x 2 y x 2 ·⎩ − yy X 1 xX 1 2 y ∣∣∣∣∣ ⎬Xyy 2 X 2 xX 2 2 y⎭ +Y yy Y x 2 Y y⎧ ∣ ∣⎫ ⎨ ∣∣∣∣∣ X 1+ y x 2 y x 2 y x 2 ·⎩ − xX 1 1 yy Xy1 ∣∣∣∣∣ ⎬X 2 xX 2 1 yy Xy2 ⎭ .Y x 1 Y yy Y yX 1 x 1 X 1 x 2 yX 1 yX 2 x 1 X 2 x 2 yX 2 yY x 1 Y x 2 y Y y∣ ∣∣∣∣∣⎫⎬⎭ +3672.24. Appropriate formalism. To describe the combinatorics underlyingformulas (2.10), (2.22) and (2.23), as in [Me2004], let us introduce the followingnotation for the Jacobian determinant:(2.25) ∆(x 1 |x 2 |y) :=∣∣X 1 xX 1 1 xX 1 2 y ∣∣∣∣∣X 2 xX 2 1 xX 2 2 y .Y x 1 Y x 2 Y yHere, in the notation ∆(x 1 |x 2 |y), the three spaces between the two verticallines | refer to the three columns of the Jacobian determinant, and the termsx 1 , x 2 , y in (x 1 |x 2 |y) designate the partial derivatives appearing in eachcolumn. Accordingly, in the following two examples of modified Jacobiandeterminants:(2.26)⎧∆(x 1 x 2 |x 2 |y) :=⎪⎨∣∆(x 1 |x 2 |x 1 y) :=⎪⎩∣X 1 x 1 x 2 X 1 x 2 Xy1X 2 x 1 x 2 X 2 x 2 Xy2Y x 1 x 2 Y x 2 Y yX 1 x 1 X 1 x 2 X 1 x 1 yX 2 x 1 X 2 x 2 X 2 x 1 yY x 1 Y x 2 Y x 1 y∣,∣andwe simply mean which column of first order derivatives is replaced by acolumn of second order derivatives in the original Jacobian determinant.As there are 6 possible second order derivatives (·) x 1 x 1, (·) x 1 x 2, (·) x 1 x y,(·) x 2 x 2, (·) x 2 y and (·) yy together with 3 columns, we obtain 3 × 6 = 18
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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 (
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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 + [
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
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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 and 348: Appying the chain rule, we may comp
Page 349 and 350: Secondly:(5.3)Y j i 1 ,i 2= Y jx i
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: eplacing the third column by the se
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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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