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

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208Proof. Reading the three formulas just preceding, by adding the first one tothe second one multiplied by y x = Q x together with the third one multipliedby F = Q xx , one visibly sees that the coefficients of both ∂ ∂a and ∂ ∂b dovanish in the obtained sum, as announced.Keeping in mind — so as to avoid any confusion — that the same letter xis used to denote simultaneously the independent variable of the differentialequation y xx = F(x, y, y x ) and the non-parameter variable of the associatedsubmanifold of solutions y = Q(x, a, b), we may now write this two-waystransfer D ←→ ∂ x exactly as we did in the above three equations, namelysimply as an equality between two derivations living in the (x, y, y x )-spaceand in the (x, a, b)-space:D = ∂ x .Lemma. With G = G(x, y, y x ) being any local K-analytic function in the(x, y, y x )-space, the three second-order derivatives G yxy x, G yyx and G yy expressas follows in terms of the second-order jet Jx,a,b 2 (T) of the definingfunction T :G yxy x= Q b Q b∆ 2+ T a∆ 3 (+ T b∆ 3 (G yyx = − Q b Q xb∆ 2+ T (a∆ 3+ T b∆ 3 (G yy = Q xb Q xb∆ 2+ T (a∆ 3+ T b∆ 3 (T aa − 2 Q a Q b∆ 2 T ab + Q a Q a∆ 2 T bb +∣ ∣ ∣ ∣ ∣ ∣)∣∣∣ QQ a Q b Q bb ∣∣∣ ∣∣∣ Qa − 2 QQ xb Q a Q b Q ab ∣∣∣ ∣∣∣ Qb + Qxbb Q xb Q b Q b Q aa ∣∣∣b +xab Q xb Q xaa∣ ∣ ∣ ∣ ∣ ∣)∣∣∣ Q− Q a Q a Q bb ∣∣∣ ∣∣∣ Qa + 2 QQ xa Q a Q a Q ab ∣∣∣ ∣∣∣ Qb − Qxbb Q xa Q b Q a Q aa ∣∣∣bxab Q xa Q xaaT aa + Q a Q xb + Q b Q xa∆ 2T ab − Q a Q xa∆ 2 T bb +∣ ∣ ∣∣∣ Q− Q a Q b Q bb ∣∣∣xa + ( ) ∣ ∣ ∣ ∣)QQ xb Q a Q xb + Q b Q xa ∣∣ Q b Q ab ∣∣∣ ∣∣∣ Q− Qxbb Q xb Q b Q b Q aa ∣∣∣xb +xab Q xb Q xaa∣ ∣ ∣∣∣ QQ a Q a Q bb ∣∣∣xa − ( ) ∣ ∣ ∣ ∣)QQ xa Q a Q xb + Q b Q xa ∣∣ Q a Q ab ∣∣∣ ∣∣∣ Q+ Qxbb Q xa Q b Q a Q aa ∣∣∣xbxab Q xa Q xaaT aa − 2 Q xa Q xb∆ 2T ab + Q xa Q xa∆ 2 T bb +∣ ∣ ∣ ∣ ∣ ∣)∣∣∣ QQ xa Q b Q bb ∣∣∣ ∣∣∣ Qxa − 2 QQ xb Q xa Q b Q ab ∣∣∣ ∣∣∣ Qxb + Qxbb Q xb Q xb Q b Q aa ∣∣∣xb +xab Q xb Q xaa∣ ∣ ∣ ∣ ∣ ∣)∣∣∣ Q− Q xa Q a Q bb ∣∣∣ ∣∣∣ Qxa + 2 QQ xa Q xa Q a Q ab ∣∣∣ ∣∣∣ Qxb − Qxbb Q xa Q xb Q a Q aa ∣∣∣xb .xab Q xa Q xaaProof. We apply the operatororder derivative G yx , namely we consider:∂∂y x, wiewed in the (x, a, b)-space, to the first∂ yx(Gyx)=∂∂y x[− Q b∆ T a + Q a∆ T b],
and we then expand carefully the result by collecting somewhat in advancethe obtained terms with respect to the derivatives of T :(G yxy x= − Q b ∂∆ ∂a + Q )[a ∂− Q b∆ ∂b ∆ T a + Q ]a∆ T b(Qb Q ab∆ − Q )b Q b ∆ a∆ ∆ 2 T a + Q b Q b∆ ∆ T aa+=∆(+ − Q b∆(+ − Q a∆(Qa+∆Q aa∆ + Q )b Q a ∆ a∆ ∆ 2 T b − Q b∆Q bb∆ + Q )a Q b ∆ b∆ ∆ 2 T a − Q a∆Q ab∆ − Q a∆)Q a ∆ b∆ 2 T b + Q a∆Q a∆ T ab+Q b∆ T ab+Q a∆ T bb.The terms involving T aa , T ab , T bb are exactly the ones exhibited by the lemmafor the expression of G yxy x. In the four large parentheses which are coefficientsof T a , T b , T a , T b , we replace the occurences of ∆ a , ∆ a , ∆ b , ∆ b simplyby:∆ a = Q xb Q aa + Q a Q xab − Q xa Q ab − Q b Q xaa∆ b = Q xb Q ab + Q a Q xbb − Q xa Q bb − Q b Q xab ,and the total sum of terms coefficiented by T a in our expression now becomes:T(a [ ] [ ]∆ 3 Q b Q ab Qa Q xb − Q b Q xa − Qb Q b Qxb Q aa + Q a Q xab − Q xa Q ab − Q b Q xaa −[ ] [ ] )− Q a Q bb Qa Q xb − Q b Q xa + Qa Q b Qxb Q ab + Q a Q xbb − Q xa Q bb − Q b Q xab == T (a∆ 3 Q a Q b Q xb Q ab − Q b Q b Q xa Q ab1 − Q b Q b Q xb Q aa − Q a Q b Q b Q xab ++ Q b Q b Q xa Q ab1 + Q b Q b Q b Q xaa −− Q a Q a Q xb Q bb + Q a Q b Q xa Q bb2 + Q a Q b Q xb Q ab + Q a Q a Q b Q xbb −)− Q a Q b Q xa Q bb − Q 2a Q b Q b Q xab == T (a [ ] [ ]∆ 3 Q a Q a Qb Q xbb − Q xb Q bb − 2 Qa Q b Qb Q xab − Q xb Q ab ++ Q b Q b[Qb Q xaa − Q xb Q aa] ) ,so that we now have effectively reconstituted the three 2 × 2 determinantsappearing in the second line of the expression claimed by the lemma forthe transfer of G yxy xto the (x, a, b)-space. The treatment of the coefficientof T bmakes only a few differences, hence will be skipped here (but not∆ 3in the manuscript). Finally, the two remaining expressions for G yyx andfor G yy are obtained by performing entirely analogous algebrico-differentialcomputations.End of the proof of the Main Theorem. Applying the above formula forG yxy xwith x := z, with a := z, with b := w, with ∆ := Θ z Θ zw − Θ w Θ zz ,209
Page 1 and 2:
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 .
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 and 162: Observe that these expressions are
Page 163 and 164: where the first term I involves onl
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: Then thanks to a straightforward ap
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 and 220: then by replacing the(so obtained)v
Page 221 and 222: and also in addition the fact that
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Page 225 and 226: the leaves of this local foliation
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
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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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