Travaux sur les symétries de Lie des équations aux ... - DMA - Ens

176symmetries of (E 0 ) have the following form:(7.28) ( (αl,0+ ∑ nk=1(x, u) ↦−→α l,k x k1 + ∑ nk=1 ε ,k x k)1≤l≤n(∑ mi 1=1 βj i 1u i1 + γ 0,j + ∑ nk 1=1 γ1,j k 1x k1 + · · · + ∑ ⎞k 1≤···≤k κ−1γ κ−1,jk 1,...,k κ−1x k1 · · · x kκ−1[1 + ∑ nk=1 ε ⎠.k x k ])1≤j≤mκ−1We note again that these transformations preserve the solutions of (E 0 ) :u j x k1···x kκ= 0, namely the graphs of polynomial maps of degree ≤ (κ − 1)from K n to K m .5.3. Nonhomogeneous system. Let κ ≥ 3. Let us expand the definingequations (7.28) as done in (7.28). We will write only the coefficientsof the five monomial families ct., U i 1l 1 ,...,l κ−2, U i 1l 1 ,...,l κ−1, U i 1l 1U i 2l 2 ,...,l κandU i 1l 1 ,l 2U i 2l 3 ,...,l κ+1. Moreover, we fix always l 1 = l 2 = · · · = l κ = l κ+1 = land i 1 = i 2 , except for the fourth family of monomials where we distinguishthe two cases i 1 = i 2 and i 1 ≠ i 2 . Thus we obtain six linear equations ofpartial derivatives, the members on the left side (coming from the expressionof R j 1k1 ,...,k κgiven by Lemma 8.1) coincide with the members on theright hand side of (7.28). Furthermore, the members on the right hand sideare exactly the same as those obtained in (7.28), with more indices! We usethe letters l ′ , k 1 ′ , . . ., k′ κ = 1, . . .,n and j′ , i ′ 1 = 1, . . ., m for the indices ofthe arguments of the expressions Π, obtaining the six following equations,which generalize the equations (7.28):(7.28)[1] : R j x k1 x k2 ···x kκ= Π(x, u, Q l′ , Q l′x k ′1, R j′ , R j′x k ′1, . . .,R j′x k ′ 1···x k ′κ−1, R j′u i′ 1),([1] : Rx j k1 x k2···x kκ= Π x, u, Q l′ , Q l′x k, R j′ , R′ x j′k, . . .,R′ x j′k ′1 1 1···x k, R j′′κ−1∑[2] : δ l,.........,lk σ(1) ,...,k σ(κ−2)R j −x kσ(κ−1) x kσ(κ) u i 1σ∈S κ−2κ− δ j i 1⎛⎝ ∑(= Π x, u, Q l′ , Q l′x k, Q l′′1σ∈S κ−3κδ l,.........,lk σ(1) ,...,k σ(κ−3)Q l x kσ(κ−2) x kσ(κ−1) x kσ(k)⎞⎠ =x k ′ x k, R j′ , R′ x j′k, . . .,R′ x j′k ′1 2 1 1···x k, R j′′κ−1u i′ 1u i′ 1), R j′.x k ′ u i′ 11).